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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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✓ 27 results · all verified · 19 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Mackeys Imprimitivity Theorem

1 · Prerequisites

2 · Summary

Mackey's imprimitivity theorem classifies the transitive systems of imprimitivity of a second-countable locally compact group $G$ on a homogeneous space $G / H$ : they are exactly the systems induced from a strongly continuous unitary representation $σ$ of the closed subgroup $H$ , and the classification is a bijection of unitary equivalence classes. The page builds the machinery from the definition of a system of imprimitivity and its transformation (covariance) algebra, through the $\hat{N}$ -spectral measure of a representation of an abelian normal subgroup, the multiplicity model of a projection-valued measure over a standard Borel base, Borel cross-sections and Haar lifts on $G / H$ , and the regularization of transitive unitary cocycles by the stabilizer representation. It ends with the imprimitivity theorem, its uniqueness clause, and the little-group reduction for an abelian normal subgroup with regular dual orbits, which expresses every irreducible representation of a topological semidirect product as an induction from irreducible stabilizer data. The standing hypothesis is the Axiom of Choice, used through the rho-function, Fubini, Bochner and spectral-multiplicity suppliers.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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Systems of imprimitivity for a Borel $G$ -space

Definition

Let $G$ be a second-countable locally compact Hausdorff topological group (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Second countability: an at most countable basis for the topology, Topological group: multiplication and inversion are continuous) acting measurably on a standard Borel space $X$ (Standard Borel spaces), that is, the map $(g, x) \mapsto g x$ is $B (G) \otimes B (X)$ -measurable (Left group actions, transitive actions, and faithful actions, A measurable function between measurable spaces, Measurable spaces and measurable sets), and let $H$ be a separable complex Hilbert space (Hilbert space, Separability: the existence of an at most countable dense subset). A system of imprimitivity for the action is a pair $(U, P)$ in which $U : G \to U (H)$ is a strongly continuous unitary representation (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) and $P$ is a projection-valued measure on the Borel $σ$ -algebra of $X$ (Projection valued measure) such that

$U_{g} P (E) U_{g}^{- 1} = P (g E) (g \in G, E \subseteq X Borel) .$

The family of $P$ -null Borel sets is the null-set class of the system; the system is ergodic when every Borel $E$ with $U_{g} P (E) U_{g}^{- 1} = P (E)$ for all $g \in G$ satisfies $P (E) = 0$ or $P (E) = I$ , and is nonzero when $P (X) = I$ and $H \neq {0}$ .

Well-definedness. The covariance relation is a condition on the given pair: for fixed $g$ the map $E \mapsto P (g E)$ is a projection-valued measure because $E \mapsto g E$ is a $σ$ -algebra automorphism of $B (X)$ , and $U_{g} P (E) U_{g}^{- 1}$ is the projection $U_{g} (P (E) H)$ with the same range as $P (E)$ transported by the unitary $U_{g}$ , so both sides of the displayed identity are orthogonal projections (Projection valued measure); since $U_{g}$ is unitary, $U_{g}^{- 1} = U_{g}^{*}$ throughout. The null-set class is a $σ$ -ideal of $B (X)$ : a projection $P (E)$ vanishes exactly when the finite scalar set functions $E \mapsto ⟨ P (E) ξ, ξ ⟩$ ( $ξ \in H$ ) all vanish, and these are countably additive because the series in clause 4 of the projection-valued measure definition converges in norm and the inner product is continuous (Projection valued measure). No regularity of $P$ is assumed, no topological condition beyond measurability of the action is imposed, and the definition itself makes no choice.

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The transformation (covariance) algebra $C_{c} (G \times X)$

Definition

Assume AC (The Axiom of Choice). Let $G$ be a locally compact Hausdorff topological group (Topological group: multiplication and inversion are continuous, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space) acting continuously on a locally compact Hausdorff space $X$ (Left group actions, transitive actions, and faithful actions, Continuity of a map of topological spaces at a point and globally, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space), with a fixed left Haar measure $d g$ (Left Haar integral and left Haar measure) and modular function $Δ_{G}$ (Modular function of a locally compact group). The transformation algebra of the action is the complex vector space $C_{c} (G \times X)$ of continuous complex functions with compact support (Compact support, $C_{c} (X)$ , and $C_{0} (X)$ ), equipped with the twisted convolution

$(f_{1} * f_{2}) (g, x) = \int_{G} f_{1} (h, x) f_{2} (h^{- 1} g, h^{- 1} x) d h$

and the involution

$f^{*} (g, x) = Δ_{G} (g)^{- 1} \overline{f (g^{- 1}, g^{- 1} x)} .$

Well-definedness: support and continuity. Let $K_{i} \subseteq G$ , $L_{i} \subseteq X$ be compact sets with $supp f_{i} \subseteq K_{i} \times L_{i}$ ( $i = 1, 2$ ). If $f_{2} (h^{- 1} g, h^{- 1} x) \neq 0$ then $h \in g K_{2}^{- 1}$ and $h^{- 1} x \in L_{2}$ , while $f_{1} (h, x) \neq 0$ forces $h \in K_{1}$ ; hence the integrand of $(f_{1} * f_{2}) (g, x)$ is supported in the compact set $K_{1} \cap g K_{2}^{- 1}$ , which is nonempty only for $g \in K_{1} K_{2}$ , and the integral is a finite number by finiteness of Haar measure on compacta. For $g$ in a compact neighbourhood $V$ of a fixed $g_{0}$ the $h$ -support lies in the fixed compact set $K = K_{1} \cap V K_{2}^{- 1}$ , and $x$ in a compact neighbourhood of a fixed $x_{0}$ ; the map $(h, g, x) \mapsto f_{1} (h, x) f_{2} (h^{- 1} g, h^{- 1} x)$ is continuous on $G \times G \times X$ as a composition of the continuous group operations and the continuous action (Topological group: multiplication and inversion are continuous, Left group actions, transitive actions, and faithful actions), so it is uniformly continuous on the compact set $K \times \overline{V} \times L$ . Given $ε > 0$ there is a neighbourhood $U$ of $(g_{0}, x_{0})$ with $∣ f_{1} (h, x) f_{2} (h^{- 1} g, h^{- 1} x) - f_{1} (h, x_{0}) f_{2} (h^{- 1} g_{0}, h^{- 1} x_{0}) ∣ \leq ε$ for all $h \in K$ and $(g, x) \in U$ ; both integrands vanish off $K$ , so $∣ (f_{1} * f_{2}) (g, x) - (f_{1} * f_{2}) (g_{0}, x_{0}) ∣ \leq ε ∣ K ∣$ , where $∣ K ∣$ is the finite Haar measure of $K$ . This proves continuity of $f_{1} * f_{2}$ ; its support is contained in $(K_{1} K_{2}) \times L_{1}$ , a compact set, so $f_{1} * f_{2} \in C_{c} (G \times X)$ . The product is bilinear in $(f_{1}, f_{2})$ by linearity of the Haar integral.

Well-definedness: associativity. Fix $(g, x)$ and put $F (h, r) = f_{1} (h, x) f_{2} (h^{- 1} r, h^{- 1} x) f_{3} (r^{- 1} g, r^{- 1} x)$ ; it is continuous and compactly supported in $(h, r)$ , with support in the compact set $K_{1} \times (K_{1} K_{2} \cap g K_{3}^{- 1})$ by the support computation above. Writing each convolution as its defining integral, the left-hand side of $(f_{1} * f_{2}) * f_{3} = f_{1} * (f_{2} * f_{3})$ at $(g, x)$ is the iterated integral $\int_{G} \int_{G} F (h, r) d h d r$ , while the right-hand side is $\int_{G} \int_{G} F (h, h k) d k d h$ ; by Compactly supported kernels admit commuting radon integrals applied to the continuous compactly supported kernel $F$ the order of the first iterated integral may be interchanged, and the inner substitution $r = h k$ , which preserves the left Haar integral by Left Haar integral and left Haar measure and changes $h^{- 1} r \mapsto k$ , $r^{- 1} g \mapsto k^{- 1} h^{- 1} g$ , $r^{- 1} x \mapsto k^{- 1} h^{- 1} x$ , identifies them. Hence $*$ is associative.

Well-definedness: involution. The function $f^{*}$ is continuous, since $(g, x) \mapsto (g^{- 1}, g^{- 1} x)$ and $Δ_{G}$ are continuous (Modular function of a locally compact group, The modular function is a continuous homomorphism), and its support is the image of the compact set $supp f$ under that homeomorphism, hence compact; so $f^{*} \in C_{c} (G \times X)$ . Applying $*$ twice and using that $Δ_{G}$ is a continuous homomorphism into the positive reals, so that $Δ_{G} (g^{- 1}) = Δ_{G} (g)^{- 1}$ , gives $(f^{*})^{*} (g, x) = Δ_{G} (g)^{- 1} \overline{Δ_{G} (g^{- 1})^{- 1} \overline{f (g, x)}} = f (g, x),$ that is, $(f^{*})^{*} = f$ . In the same way, for the products one computes $(f_{1} * f_{2})^{*} (g, x) = Δ_{G} (g)^{- 1} \int_{G} \overline{f_{1} (h, g^{- 1} x)} \overline{f_{2} (h^{- 1} g^{- 1}, h^{- 1} g^{- 1} x)} d h,$ and substituting $h = g ℓ$ in the defining integral of $(f_{2}^{*} * f_{1}^{*}) (g, x)$ turns its modular factor into $Δ_{G} (g)^{- 1}$ because $Δ_{G} (g ℓ)^{- 1} Δ_{G} (ℓ^{- 1})^{- 1} = Δ_{G} (g)^{- 1}$ , so $(f_{1} * f_{2})^{*} = f_{2}^{*} * f_{1}^{*}$ . Together with the conjugate-linearity of $*$ this says that $C_{c} (G \times X)$ with $*$ and $^{*}$ is a complex associative algebra with involution; the involution of the group convolution is the special case of the published definition (Compactly supported convolution on a group).

Trivial action. If $g x = x$ for all $g, x$ , then the twisted product becomes $(f_{1} * f_{2}) (g, x) = \int_{G} f_{1} (h, x) f_{2} (h^{- 1} g, x) d h$ , which is convolution in the group variable with pointwise multiplication in the base variable, with the factor order of Compactly supported convolution on a group.

AC is inherited through Compactly supported kernels admit commuting radon integrals, which commutes the two Radon integrals in the associativity computation; no independent choice step is used, and the support, continuity and involution computations themselves make no choice.

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Characters of the L1 algebra of an abelian group

Statement

Assume AC. For a second-countable LCH abelian group $N$ with Haar measure, every nonzero complex-linear multiplicative functional $λ$ on $L^{1} (N)$ is uniquely $λ (f) = \int_{N} f (n) χ (n) d n$ for a continuous unitary character $χ$ . This bijection from $\hat{N}$ with its compact-open topology to the character space with its pointwise-evaluation topology is a homeomorphism. No Pontryagin duality or Fourier inversion theorem is assumed.

Facts & Assumptions

Given: AC, a second-countable LCH abelian group $N$ with a fixed left Haar measure $μ$ , and a nonzero complex-linear multiplicative functional $λ : L^{1} (N) \to C$ .

[F1]

$L^{1} (N)$ is a complex Banach $*$ -algebra whose convolution is bilinear, associative and contractive, $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1}$ , and agrees with the $C_{c}$ convolution $u * v (x) = \int_{N} u (y) v (y^{- 1} x) d y$ whenever both arguments lie in $C_{c} (N)$ (L1 of a locally compact group is a Banach star-algebra, Convolution on L1 of a locally compact group).

[F2]

$L^{1} (N)$ is complete and $C_{c} (N)$ is dense in it (Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F3]

For $n \in N$ the translation operator $L_{n} f (x) = f (n^{- 1} x)$ is linear and isometric on $L^{1} (N)$ , $L_{n} L_{m} = L_{n m}$ , and $n \mapsto L_{n} f$ is continuous in the norm of $L^{1} (N)$ for every $f$ (Strong continuity of left and modular right translations on L1 and L2).

[F4]

A character of a nonzero unital complex Banach algebra is unital and satisfies $∣ χ (a) ∣ \leq ∥ a ∥$ (Characters on a unital Banach algebra are continuous).

[F5]

A strongly measurable Banach-valued function with $\int ∥ g ∥ d μ < \infty$ is Bochner integrable, and its integral obeys $∥ \int g ∥ \leq \int ∥ g ∥$ ; a bounded linear $T$ commutes with the Bochner integral, $T (\int g) = \int T g$ (Bochner integrability criterion, Bochner integral norm inequality, Bounded linear maps commute with Bochner integration, Strongly measurable Banach-valued function).

[F6]

For $σ$ -finite measure spaces $(X, μ), (Y, ν)$ and $h \in L^{1} (μ \times ν)$ the two iterated integrals agree with the product integral; the compactly supported instances used below satisfy the $σ$ -finiteness hypothesis because on a compact subset of $N \times N$ the restricted Haar measures are finite (Fubini's theorem for L^1 functions on a sigma-finite product, Compactly supported kernels admit commuting radon integrals).

[F7]

Haar measure is positive on nonempty open sets and finite on compact sets, and compact sets admit nonnegative compactly supported cutoffs equal to one on them (Haar measure is positive on nonempty open sets and finite on compact sets, LCH Urysohn cutoff).

[F8]

$\hat{N}$ is the group of continuous homomorphisms $χ : N \to T$ with the compact-open topology, and the character space of $L^{1} (N)$ carries the topology of pointwise evaluation (The Pontryagin dual with the compact-open topology, Character and maximal ideal space).

[F9]

AC is the standing hypothesis (The Axiom of Choice).

Proof

technique · direct

Given: AC, a second-countable LCH abelian group $N$ with left Haar measure, and a nonzero complex-linear multiplicative $λ$ on $L^{1} (N)$ .

1.1F1F4F9algebra

Put $\tilde{A} = C \oplus L^{1} (N)$ with $(z, f) (w, g) = (z w, z g + w f + f * g)$ and $∥ (z, f) ∥ = ∣ z ∣ + ∥ f ∥_{1}$ . The product is bilinear and associative, $∥ (z, f) (w, g) ∥ \leq (∣ z ∣ + ∥ f ∥_{1}) (∣ w ∣ + ∥ g ∥_{1})$ , and $\tilde{A}$ is complete, so it is a nonzero unital complex Banach algebra with unit $(1, 0)$ ; the map $\tilde{λ} (z, f) = z + λ (f)$ is complex-linear, multiplicative because $λ$ is multiplicative, and $\tilde{λ} (1, 0) = 1$ , so it is a character. By [F4], $∣ λ (f) ∣ = ∣ \tilde{λ} (0, f) ∣ \leq ∥ (0, f) ∥ = ∥ f ∥_{1}$ for every $f$ .

1.2F3choose

Since $λ \neq 0$ there is $k \in L^{1} (N)$ with $λ (k) \neq 0$ ; fix such a $k$ and set $χ (n) = λ (L_{n} k) / λ (k)$ for $n \in N$ .

1.3F1F2F3algebra

For all $f, k \in L^{1} (N)$ and all $n \in N$ one has $(L_{n} f) * k = f * (L_{n} k)$ : for $f, k \in C_{c} (N)$ both sides are continuous functions computed by the pointwise convolution formula, and substituting $y = n z$ in $\int_{N} f (n^{- 1} y) k (y^{- 1} x) d y$ uses left invariance of $d y$ to give $\int_{N} f (z) k (z^{- 1} n^{- 1} x) d z = (f * L_{n} k) (x)$ ; both sides are bounded bilinear in $(f, k)$ by [F1] and [F3], and $C_{c} (N) \times C_{c} (N)$ is dense in $L^{1} (N) \times L^{1} (N)$ by [F2], so the identity extends to all $f, k \in L^{1} (N)$ .

1.4F1F2F5F6algebra

For all $f, k \in L^{1} (N)$ , $f * k = \int_{N} f (n) L_{n} k d n$ as a Bochner integral. Approximate $f$ in $L^{1}$ by $u_{m} \in C_{c} (N)$ and pass to a subsequence with $u_{m} \to f$ a.e.; each $n \mapsto u_{m} (n) L_{n} k$ is continuous and compactly supported hence strongly measurable, and a diagonal selection of their defining simple approximants shows that the a.e. limit $n \mapsto f (n) L_{n} k$ is strongly measurable; since $\int_{N} ∥ f (n) L_{n} k ∥_{1} d n = ∥ f ∥_{1} ∥ k ∥_{1} < \infty$ , it is Bochner integrable by [F5]. The assignment $f \mapsto \int_{N} f (n) L_{n} k d n$ is bounded linear, and for $f, k \in C_{c} (N)$ pairing with any $φ \in L^{\infty} (N)$ and commuting the bounded functional through the Bochner integral reduces the identity to $\int_{N} \int_{N} f (n) k (n^{- 1} x) φ (x) d x d n = \int_{N} (f * k) (x) φ (x) d x$ , which follows from [F6] because the kernel is compactly supported; the pairing with all of $L^{\infty} (N)$ separates points of $L^{1} (N)$ , and both sides are bounded linear in $f$ with $C_{c} (N)$ dense by [F2], so the identity holds for all $f \in L^{1} (N)$ ; repeating the same density argument in the second variable gives it for all $k$ as well.

1.5F1F2F6F7algebra

Conversely, for a continuous character $χ$ define $λ_{χ} (f) = \int_{N} f χ d n$ . Then $λ_{χ}$ is complex-linear with $∣ λ_{χ} (f) ∣ \leq ∥ f ∥_{1}$ , and it is multiplicative: for $f, g \in C_{c} (N)$ the double integral $\int_{N} \int_{N} f (y) g (y^{- 1} x) χ (x) d y d x$ equals by [F6] the iterated integral $\int_{N} f (y) \int_{N} g (z) χ (y z) d z d y = λ_{χ} (f) λ_{χ} (g)$ after the substitution $z = y^{- 1} x$ and using $χ (y z) = χ (y) χ (z)$ ; both $λ_{χ} (f * g)$ and $λ_{χ} (f) λ_{χ} (g)$ are bounded bilinear in $(f, g)$ , so density of $C_{c} (N)$ ([F2]) extends multiplicativity to all $f, g \in L^{1} (N)$ . And $λ_{χ} \neq 0$ : by continuity of $χ$ at $e$ there is a nonempty open set $U$ with $Re χ > 1 / 2$ on $U$ , and by [F7] there is $c \in C_{c} (N)$ with $c \geq 0$ , $c \neq 0$ , supported in $U$ ; then $Re λ_{χ} (c) = \int_{N} c Re χ d n > 0$ , so $λ_{χ} (c) \neq 0$ .

2.1step 1.2step 1.3

Multiplicativity of $λ$ applied to [step 1.3] with this $k$ gives $λ (L_{n} f) λ (k) = λ (f) λ (L_{n} k)$ , hence $λ (L_{n} f) = χ (n) λ (f)$ for every $f \in L^{1} (N)$ and every $n \in N$ .

2.2F2F7F8step 1.5

If $χ_{i} \to χ$ in the compact-open topology, then $λ_{χ_{i}} (f) \to λ_{χ} (f)$ for every $f \in L^{1} (N)$ : given $ε > 0$ choose $u \in C_{c} (N)$ with $∥ f - u ∥_{1} < ε / 4$ ([F2]); then $∣ λ_{χ_{i}} (f) - λ_{χ} (f) ∣ \leq 2 ∥ f - u ∥_{1} + ∥ u ∥_{\infty} \int_{supp u} ∣ χ_{i} - χ ∣ d n$ , and $χ_{i} \to χ$ uniformly on the compact set $supp u$ directly from the compact-open subbasis, while the Haar measure of $supp u$ is finite by [F7]. Thus the map $χ \mapsto λ_{χ}$ is continuous for the two stated topologies.

3.1step 1.2step 2.1F3

$χ$ is multiplicative: since $L_{n} L_{m} = L_{n m}$ and $λ (k) \neq 0$ , applying [step 2.1] to $L_{m} k$ gives $χ (n m) λ (k) = λ (L_{n} L_{m} k) = χ (n) λ (L_{m} k) = χ (n) χ (m) λ (k)$ , so $χ (n m) = χ (n) χ (m)$ ; in particular $χ (e) = 1$ and $χ (n^{- 1}) = χ (n)^{- 1}$ .

3.2step 1.1step 2.1F3

$χ$ is continuous: for $n \to n_{0}$ in $N$ one has $∣ χ (n) - χ (n_{0}) ∣ = ∣ λ (L_{n} k - L_{n_{0}} k) ∣ / ∣ λ (k) ∣ \leq ∥ L_{n} k - L_{n_{0}} k ∥_{1} / ∣ λ (k) ∣ \to 0$ by [step 1.1] and [F3].

3.3step 1.1step 2.1step 1.4F5

Apply the bounded functional $λ$ to [step 1.4] and commute it through the Bochner integral: $λ (f) λ (k) = λ (f * k) = \int_{N} f (n) λ (L_{n} k) d n = λ (k) \int_{N} f (n) χ (n) d n$ by [step 2.1]; since $λ (k) \neq 0$ , dividing gives the classification formula $λ (f) = \int_{N} f (n) χ (n) d n$ for every $f \in L^{1} (N)$ .

3.4step 1.1step 1.2step 2.1F3F8algebra

Conversely, suppose $λ_{i} \to λ$ in the pointwise-evaluation topology of the character space. Fix the $k$ of [step 1.2] and a compact $C \subseteq N$ . The set ${L_{n} k : n \in C}$ is norm compact in $L^{1} (N)$ as the continuous image of $C$ under [F3], so for each $ε > 0$ it has a finite $ε / 3$ -net $L_{n_{1}} k, \dots, L_{n_{r}} k$ . For all sufficiently large $i$ one has $∣ λ_{i} (L_{n_{j}} k) - λ (L_{n_{j}} k) ∣ < ε / 3$ for every $j$ and $∣ λ_{i} (k) - λ (k) ∣ < \min {ε, ∣ λ (k) ∣ / 2}$ , using [step 1.1] for the bounds $∥ λ_{i} ∥ \leq 1$ and $∥ λ ∥ \leq 1$ ; then for every $n \in C$ and the corresponding $j$ one gets $∣ λ_{i} (L_{n} k) - λ (L_{n} k) ∣ < ε$ , and division by the eventually nonvanishing $λ_{i} (k)$ gives $∣ χ_{i} (n) - χ (n) ∣ \leq M ε$ uniformly on $C$ for a constant $M$ depending only on $λ (k)$ and $∥ k ∥_{1}$ . Hence $λ_{i} \to λ$ pointwise implies $χ_{i} \to χ$ uniformly on compacta, that is, the inverse map is continuous.

4.1step 3.1step 3.2algebra

$∣ χ (n) ∣ = 1$ for every $n$ : [step 1.1] gives $∣ χ (n) ∣ \leq ∥ k ∥_{1} / ∣ λ (k) ∣$ , and applying [step 3.1] to the powers $n^{j}$ gives $∣ χ (n) ∣^{j} \leq ∥ k ∥_{1} / ∣ λ (k) ∣$ for all $j \geq 1$ , whence $∣ χ (n) ∣ \leq 1$ ; replacing $n$ by $n^{- 1}$ and using $χ (n^{- 1}) = χ (n)^{- 1}$ gives $∣ χ (n) ∣ \geq 1$ as well. Thus $χ : N \to T$ is a continuous character.

5.1step 1.5step 3.3step 4.1algebra

If $λ_{χ_{1}} = λ_{χ_{2}} = λ$ , choose $k$ with $λ (k) \neq 0$ . Substitution $x = n y$ in the defining integral gives $λ_{χ_{i}} (L_{n} k) = χ_{i} (n) λ_{χ_{i}} (k)$ for $i = 1, 2$ . Hence $χ_{i} (n) = λ (L_{n} k) / λ (k)$ for every $n$ , so $χ_{1} = χ_{2}$ . Together with step 3.3 this proves the bijection.

6.1step 3.3step 5.1step 2.2step 3.4∎

Steps [2.2] and [3.4] show that $χ \mapsto λ_{χ}$ is a homeomorphism from $\hat{N}$ with the compact-open topology onto the character space with the pointwise-evaluation topology, and [step 3.3] with [step 5.1] shows every nonzero complex-linear multiplicative functional is uniquely of the form $λ_{χ}$ .

Remarks

The proof uses no Pontryagin duality and no Fourier inversion: the characters are produced from $λ$ itself through the translation identity, and the only harmonic-analytic inputs are translation continuity, Haar positivity and the Bochner/Fubini calculus.

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Direct integrals transport along bimeasurable base isomorphisms

Statement

Assume AC. Let $(X, B_{X}, μ)$ and $(Y, B_{Y}, ν)$ be $σ$ -finite standard Borel measure spaces, let $c : X \to Y$ be a bimeasurable bijection with $ν = c_{*} μ$ , and let $(H_{y})_{y \in Y}$ be a measurable complex Hilbert field over $(Y, ν)$ with direct integral $\int_{Y}^{\oplus} H_{y} d ν (y)$ . Then $x \mapsto H_{c (x)}$ is a measurable Hilbert field over $(X, μ)$ with the pulled-back fundamental family, and pullback of sections $ξ \mapsto ξ \circ c$ is a unitary $c^{*} : \int_{Y}^{\oplus} H_{y} d ν (y) ⟶ \int_{X}^{\oplus} H_{c (x)} d μ (x)$ that intertwines multiplication by $f \in L^{\infty} (Y, ν)$ with multiplication by $f \circ c$ . A decomposable operator field $(T_{y})$ over $Y$ corresponds to the decomposable field $x \mapsto T_{c (x)}$ over $X$ with the same essential norm and the same fibrewise adjoint and product identities.

Facts & Assumptions

Given: AC, $σ$ -finite standard Borel measure spaces $(X, μ)$ , $(Y, ν)$ , a bimeasurable bijection $c : X \to Y$ with $ν = c_{*} μ$ , and a measurable Hilbert field $(H_{y}, e_{n} (y))_{y \in Y}$ with countable fundamental family and direct integral $H_{Y} = \int_{Y}^{\oplus} H_{y} d ν (y)$ .

[F1]

The field datum means exactly: a separable Hilbert space $H_{y}$ for each $y$ , vectors $e_{n} (y)$ spanning a dense subspace of $H_{y}$ , Borel Gram coefficients $y \mapsto ⟨ e_{n} (y), e_{m} (y) ⟩$ ; a section is measurable when all coefficients $y \mapsto ⟨ ξ (y), e_{n} (y) ⟩$ are Borel, and sections are identified when they agree off a Borel null set (Measurable Hilbert field from a countable fundamental family).

[F2]

$H_{Y}$ is the quotient of the square-integrable measurable sections by almost-everywhere agreement, with inner product $⟨ [ξ], [η] ⟩ = \int_{Y} ⟨ ξ (y), η (y) ⟩ d ν (y)$ , and it is a Hilbert space (Direct integral of a measurable Hilbert field, Direct integrals of measurable Hilbert fields are Hilbert spaces).

[F3]

For a section, coefficient measurability is equivalent to measurability of all pairings with measurable sections; such sections are closed under measurable scalar combinations and pointwise norm limits, and $y \mapsto ∥ ξ (y) ∥$ is measurable (Measurable sections have measurable pointwise inner products).

[F4]

Precomposition with the Borel maps $c$ and $c^{- 1}$ preserves Borel measurability (Composition with a Borel measurable outer map preserves measurability, Standard Borel spaces, Measurable spaces and measurable sets).

[F5]

An operator field $(T_{y})$ is weakly measurable when its fundamental matrix coefficients are Borel; it is essentially bounded when ${ess sup}_{y} ∥ T_{y} ∥ < \infty$ , and a bounded operator on $H_{Y}$ is decomposable when it acts by such a field, $S [ξ] = [T ξ]$ (Measurable and decomposable operator fields).

Proof

technique · direct

Given: AC, the base spaces and the field of the statement, with fundamental family $(e_{n})$ over $Y$ .

1.1F1F4

Define $e_{n}^{c} (x) : = e_{n} (c (x))$ in $H_{c (x)}$ . The Gram coefficients $x \mapsto ⟨ e_{n}^{c} (x), e_{m}^{c} (x) ⟩_{H_{c (x)}} = (⟨ e_{n}, e_{m} ⟩_{H_{∙}}) \circ c (x)$ are Borel by [F1] and [F4], and for each $x$ the span of ${e_{n}^{c} (x)}$ equals the span of ${e_{n} (c (x))}$ , which is dense in $H_{c (x)}$ ; hence $x \mapsto H_{c (x)}$ with this pulled-back family is a measurable Hilbert field with countable fundamental family over $(X, μ)$ .

2.1step 1.1F3F4

A section $ξ$ over $Y$ has Borel coefficients $⟨ ξ, e_{n} ⟩$ if and only if the section $ξ \circ c$ over $X$ has Borel coefficients $⟨ ξ \circ c, e_{n}^{c} ⟩ = (⟨ ξ, e_{n} ⟩) \circ c$ : one direction is [F4], and the converse applies [F4] to $c^{- 1}$ , which is bimeasurable; by [F3] the same equivalence holds for all pairings, and $∥ ξ \circ c ∥ = ∥ ξ ∥ \circ c$ is measurable whenever $ξ$ is.

3.1step 1.1step 2.1F1

Change of variables: for every nonnegative Borel function $h$ on $Y$ , $\int_{X} h (c (x)) d μ (x) = \int_{Y} h d ν$ , because $ν = c_{*} μ$ is the pushforward; consequently for a measurable section $ξ$ one has $\int_{X} ∥ ξ (c (x)) ∥_{H_{c (x)}}^{2} d μ (x) = \int_{Y} ∥ ξ (y) ∥_{H_{y}}^{2} d ν (y)$ , so $ξ \circ c$ is square-integrable exactly when $ξ$ is.

4.1step 2.1step 3.1F2

Define $c^{*} [ξ] : = [ξ \circ c]$ on the direct integral. It is well defined on classes: if $ξ = η$ outside a Borel $ν$ -null set $N$ , then $ξ \circ c = η \circ c$ outside $c^{- 1} (N)$ , and $μ (c^{- 1} (N)) = ν (N) = 0$ ; it is complex-linear because the fibre operations are pointwise and pullback is linear; and it preserves inner products, $⟨ c^{*} [ξ], c^{*} [η] ⟩ = \int_{X} ⟨ ξ (c (x)), η (c (x)) ⟩ d μ (x) = \int_{Y} ⟨ ξ (y), η (y) ⟩ d ν (y) = ⟨ [ξ], [η] ⟩$ , by [step 3.1]. It is surjective: for a measurable square-integrable section $η$ over $X$ , the section $ξ : = η \circ c^{- 1}$ is measurable over $Y$ by [step 2.1] applied to $c^{- 1}$ and has $ξ \circ c = η$ and the same integral by [step 3.1]. Hence $c^{*}$ is a complex-linear surjective isometry between the two direct integrals, that is, a unitary.

4.2F5step 1.1step 2.1step 3.1

A weakly measurable, essentially bounded operator field $(T_{y})$ over $Y$ pulls back to the operator field $x \mapsto T_{c (x)}$ on the fibres $H_{c (x)}$ : its fundamental matrix coefficients are $(⟨ T_{∙} e_{n}, e_{m} ⟩) \circ c$ , Borel by [F4], so the pulled field is weakly measurable, and ${x : ∥ T_{c (x)} ∥ > t} = c^{- 1} ({y : ∥ T_{y} ∥ > t})$ has $μ$ -measure $ν ({y : ∥ T_{y} ∥ > t})$ , so the two operator-norm functions have the same essential supremum; moreover, for a square-integrable section $ξ$ over $Y$ , the pulled section $T_{c (\cdot)} ξ (c (\cdot)) = (T ξ) \circ c$ is the pullback of the square-integrable section $T ξ$ , so the decomposable action is transported. Fibrewise adjoint and product identities are preserved because for each $x$ the fibre operator is $T_{c (x)}$ itself, $(T_{c (x)})^{*} = (T^{*})_{c (x)}$ and $S_{c (x)} T_{c (x)} = (S T)_{c (x)}$ .

5.1step 4.1step 4.2F2algebra∎

Finally $c^{*}$ intertwines multiplication: for $f \in L^{\infty} (Y, ν)$ and a square-integrable section $ξ$ , $c^{*} (M_{f} [ξ]) = [f ξ \circ c] = [(f \circ c) (ξ \circ c)] = M_{f \circ c} (c^{*} [ξ])$ pointwise. Together with [step 4.1] and [step 4.2] this proves that $x \mapsto H_{c (x)}$ is a measurable field, that $c^{*}$ is a unitary intertwining the two multiplication algebras, and that decomposable fields transport with the same essential norm and fibrewise algebraic identities.

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Nondegenerate representations of C0 have regular PVMs

Statement

Assume AC. Let $X$ be LCH and $T : C_{0} (X) \to B (H)$ a nondegenerate star representation. Then a unique regular PVM $P$ on $X$ with $P (X) = I$ satisfies $T (f) = \int_{X} f d P$ for every $f \in C_{0} (X)$ . The zero Hilbert space has the zero PVM.

Facts & Assumptions

Given: AC, an LCH space $X$ , a complex Hilbert space $H$ , and a nondegenerate star representation $T : C_{0} (X) \to B (H)$ .

[F1]

The one-point compactification $X^{+} = X \cup {\infty}$ is compact Hausdorff, $X$ is an open subspace carrying its own topology, and $\infty$ is the point at infinity ( $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff, The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ ).

[F2]

For a nonempty compact Hausdorff $K$ and a nonzero $H$ , every unital star homomorphism $π : C (K) \to B (H)$ is $π (f) = \int f d E$ for a unique regular PVM $E$ on $K$ (Continuous functional calculus produces a regular PVM).

[F3]

For a PVM $E$ on a measurable space, the bounded Borel integral $Φ_{E}$ is linear, multiplicative, conjugation preserving and unital; and if $E$ is a PVM on $X^{+}$ with scalar measures $E_{x}$ , then $E_{x}$ is finite (Bounded borel pvm integral, Pvm integral is a star homomorphism).

[F4]

A star representation is complex-linear with $T (\overline{f}) = T (f)^{*}$ and $T (f g) = T (f) T (g)$ ; nondegeneracy means the closed linear span of $T (C_{0} (X)) H$ equals $H$ (the convention of Continuous functional calculus produces a regular PVM).

Proof

technique · direct

Given: AC, the LCH space $X$ , the Hilbert space $H$ and the nondegenerate star representation $T$ .

1.1F3

If $H = {0}$ , let $P$ be the zero PVM, $P (B) = 0$ for every Borel $B$ ; then $P (X) = I = 0$ and $\int f d P = 0 = T (f)$ for every $f$ , and it is the only PVM on $H = 0$ . So assume $H \neq {0}$ from now on.

2.1F1F4algebrastep 1.1

Extend $T$ to a unital star homomorphism $T^{+} : C (X^{+}) \to B (H)$ by $T^{+} (f) = T (f - f (\infty) 1) + f (\infty) I$ , where $f - f (\infty) 1$ is regarded as an element of $C_{0} (X)$ through the open inclusion $X \subseteq X^{+}$ : it is continuous on $X$ and tends to $0$ at $\infty$ because $f$ does. The map $f \mapsto f - f (\infty) 1$ is linear, so $T^{+}$ is linear and $T^{+} (1) = I$ ; and $T^{+}$ is multiplicative and conjugation preserving because for $f, g \in C (X^{+})$ , writing $f = f_{0} + c$ , $g = g_{0} + d$ with $c = f (\infty)$ , $d = g (\infty)$ and $f_{0}, g_{0} \in C_{0} (X)$ , one has $f g = f_{0} g_{0} + d f_{0} + c g_{0} + c d$ with $f_{0} g_{0} + d f_{0} + c g_{0} \in C_{0} (X)$ , so $T^{+} (f g) = T (f_{0}) T (g_{0}) + d T (f_{0}) + c T (g_{0}) + c d I = T^{+} (f) T^{+} (g)$ , and $T^{+} (\overline{f}) = T (\overline{f_{0}}) + \overline{c} I = T (f_{0})^{*} + \overline{c} I = T^{+} (f)^{*}$ .

3.1F1F2step 2.1

By [F2] applied to the nonempty compact Hausdorff space $X^{+}$ and the unital star homomorphism $T^{+}$ there is a unique regular PVM $E^{+}$ on $X^{+}$ with $T^{+} (h) = \int_{X^{+}} h d E^{+}$ for every $h \in C (X^{+})$ .

4.1step 3.1F3

For every $f \in C_{0} (X)$ one has $T (f) E^{+} ({\infty}) = 0$ : since $f 1_{{\infty}} = 0$ as a bounded Borel function on $X^{+}$ and the bounded integral is multiplicative, $T (f) E^{+} ({\infty}) = Φ_{E^{+}} (f) Φ_{E^{+}} (1_{{\infty}}) = Φ_{E^{+}} (0) = 0$ .

5.1step 4.1F4

Nondegeneracy forces $E^{+} ({\infty}) = 0$ : suppose $E^{+} ({\infty}) \neq 0$ and pick $ξ = E^{+} ({\infty}) ξ \neq 0$ in its range; then for every $f \in C_{0} (X)$ and $η \in H$ , $⟨ ξ, T (\overline{f}) η ⟩ = ⟨ T (f) ξ, η ⟩ = ⟨ T (f) E^{+} ({\infty}) ξ, η ⟩ = 0$ by [step 4.1], so $ξ$ is orthogonal to the linear span of $T (C_{0} (X)) H$ , which is dense by nondegeneracy; hence $ξ = 0$ , a contradiction.

6.1step 2.1step 3.1step 5.1F1

Define $P (B) : = E^{+} (B)$ for Borel $B \subseteq X$ . This is a PVM on $X$ : the Borel sets of the open subspace $X$ are exactly the traces of Borel sets of $X^{+}$ , the values are orthogonal projections with $P (\emptyset) = 0$ , $P (X) = E^{+} (X) = I - E^{+} ({\infty}) = I$ by [step 5.1], multiplicativity and countable additivity are inherited from $E^{+}$ . For $f \in C_{0} (X)$ , $T (f) = T^{+} (f) = \int_{X^{+}} f d E^{+} = \int_{X} f d P$ , since $f$ vanishes at $\infty$ and $E^{+} ({\infty}) = 0$ .

7.1step 6.1F1F3

$P$ is regular: for each $x$ , the finite measure $P_{x} (B) = E_{x}^{+} (B)$ on $X$ is the restriction of the regular Borel measure $E_{x}^{+}$ on $X^{+}$ ; inner regularity holds because each compact subset of $X$ in the subspace topology is compact in $X^{+}$ , and outer regularity holds because open subsets of $X$ are open in $X^{+}$ .

8.1step 2.1step 3.1F2step 6.1step 7.1

Uniqueness: if $P^{'}$ is any regular PVM on $X$ with $T (f) = \int_{X} f d P^{'}$ for all $f \in C_{0} (X)$ , let $\tilde{P^{'}}$ be its extension by zero at infinity, $\tilde{P^{'}} (B) : = P^{'} (B \cap X)$ for Borel $B \subseteq X^{+}$ . This is a regular PVM on $X^{+}$ : values are orthogonal projections, $\tilde{P^{'}} (X^{+}) = P^{'} (X) = I$ and $\tilde{P^{'}} ({\infty}) = 0$ , countable additivity and multiplicativity are inherited from $P^{'}$ , and its finite scalar measures are inner regular on all Borel sets, including those containing $\infty$ , by compact approximation inside $X$ . Outer regularity follows by applying inner regularity to complements in the compact space $X^{+}$ ; thus the extension is regular. For $h \in C (X^{+})$ write $h = h_{0} + c$ with $h_{0} \in C_{0} (X)$ and $c = h (\infty)$ ; then $\int h d \tilde{P^{'}} = \int h_{0} d P^{'} + c \tilde{P^{'}} (X^{+}) = T (h_{0}) + c I = T^{+} (h)$ , so $\tilde{P^{'}}$ represents $T^{+}$ and the uniqueness in [F2] gives $\tilde{P^{'}} = E^{+}$ and hence $P^{'} = P$ .

9.1step 1.1step 6.1step 7.1step 8.1∎

Thus for nonzero $H$ there is exactly one regular PVM $P$ on $X$ with $P (X) = I$ and $T (f) = \int_{X} f d P$ , namely the restriction of $E^{+}$ ; for $H = {0}$ the zero PVM is the unique one by [step 1.1]. Both cases together prove the claim.

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Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel

Statement

Assume AC. Every second-countable locally compact Hausdorff space is Polish, hence standard Borel. Consequently, if $G$ is a second-countable locally compact Hausdorff topological group and $H \leq G$ is a closed subgroup, then the homogeneous space $G / H$ with its quotient topology is Polish and the quotient Borel structure together with the left action $G \times G / H \to G / H$ is a standard Borel $G$ -space with Borel action.

Facts & Assumptions

Given: AC, a second-countable LCH space $X$ ; later a second-countable LCH group $G$ and a closed subgroup $H \leq G$ .

[F1]

For a locally compact Hausdorff space $X$ : every point and open neighbourhood admit an open $V$ with $x \in V \subseteq \overline{V} \subseteq U$ and $\overline{V}$ compact; the open sets with compact closure form a base of $X$ ; and $X$ is regular (In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular, Countably compact, Lindel"of, sequentially compact, limit point compact and $σ$ -compact spaces, and relatively compact subsets, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space).

[F2]

AC implies DC and DC implies Countable Choice (AC implies DC implies countable choice); AC implies the ultrafilter lemma, as recorded by the locally proved upper bound in the choice ledger.

[F3]

Every regular $T_{1}$ second-countable space is metrizable (Under choice, every regular $T_{1}$ second-countable space is metrizable); in particular so is $X$ (Second countability: an at most countable basis for the topology, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[F4]

Every locally compact Hausdorff space is Čech-complete (Every locally compact Hausdorff space is Čech-complete), and a metrizable space is Čech-complete if and only if it is completely metrizable (Under the ultrafilter lemma and the Axiom of Choice, a metrizable space is Čech-complete exactly when it is completely metrizable).

[F5]

For a completely metrizable space, separability is equivalent to second countability; a Polish space is a separable completely metrizable space, and a standard Borel space is a measurable space Borel isomorphic to a Polish space (For completely metrizable spaces, the separable and second-countable definitions of Polish space agree under countable choice, Polish spaces are separable completely metrizable spaces, Standard Borel spaces).

[F6]

If $H$ is closed in an LCH group $G$ , then $G / H$ with the quotient topology is locally compact Hausdorff and the quotient map $p : G \to G / H$ is open; every compact subset of $G / H$ lies in $p (K)$ for a compact $K \subseteq G$ (Compact lifts and averaging onto C_c(G/H), Left and right cosets $g H$ and $H g$ of a subgroup, The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Topological group: multiplication and inversion are continuous).

[F7]

Multiplication $G \times G \to G$ is continuous, and the left action of a group on a quotient by a subgroup is induced by it (Topological group: multiplication and inversion are continuous, Left group actions, transitive actions, and faithful actions, Left and right cosets $g H$ and $H g$ of a subgroup).

Proof

technique · direct

Given: AC; a second-countable LCH space $X$ , later a second-countable LCH group $G$ with closed subgroup $H$ .

1.1F1

Let $B$ be a countable base of $X$ and let $B_{c}$ be the members of $B$ whose closure is compact. This is a base: given $x$ and an open $U ∋ x$ , [F1] yields an open $W$ with $x \in W \subseteq \overline{W} \subseteq U$ and $\overline{W}$ compact, and then some $B \in B$ satisfies $x \in B \subseteq W$ , so $\overline{B} \subseteq \overline{W}$ is compact and $B \in B_{c}$ . Hence every point of $X$ lies in a member of $B_{c}$ , whose closure is compact, and the closures of the countably many members of $B_{c}$ cover $X$ ; thus $X$ is a countable union of compact sets.

1.2F1F3

$X$ is regular by [F1] and $T_{1}$ because it is Hausdorff, so with its countable base it is metrizable by [F3]; fix a compatible metric $d$ .

2.1step 1.2F2F4F5

$X$ is Čech-complete by [F4], and being metrizable it is completely metrizable by the equivalence in [F4]; the choice hypotheses of [F4] are DC and the ultrafilter lemma with AC, which hold by [F2] under the standing AC. Since $X$ is second countable, [F5] makes it Polish, and then standard Borel by [F5]. This proves the first assertion.

3.1step 2.1F6

Now let $G$ be a second-countable LCH group and $H \leq G$ closed. By [F6] the quotient $G / H$ is locally compact Hausdorff and $p$ is open, so the images $p (B)$ of the members of a countable base $B$ of $G$ form a countable family of open sets; it is a base of $G / H$ because for $x \in G / H$ and an open $U ∋ x$ the preimage $p^{- 1} (U)$ is open and contains a basic $B$ through some point of the fibre, whence $x \in p (B) \subseteq U$ . Thus $G / H$ is second-countable LCH, and [step 2.1] applied to $G / H$ shows that $G / H$ is Polish and its Borel structure is standard Borel.

4.1step 3.1F7

The left action $a : G \times G / H \to G / H$ , $a (g, x H) = g x H$ , is continuous: the composite $(g, x) \mapsto p (g x)$ is continuous on $G \times G$ by [F7], it factors through the surjective open map $id \times p : G \times G \to G \times G / H$ (because $x H = x^{'} H$ implies $g x H = g x^{'} H$ ), and a continuous open surjection is a quotient map, so $a$ is continuous; in particular $a$ is Borel for the product of the Borel structures.

5.1step 2.1step 3.1step 4.1∎

Combining the two parts: every second-countable LCH space is Polish and standard Borel, and for a second-countable LCH group $G$ with closed subgroup $H$ the homogeneous space $G / H$ is Polish with standard Borel structure and the left action is continuous and hence Borel. The empty space is Polish and standard Borel by the same definitions, consistently with the vacuous case of the first assertion.

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Steinhaus and Pettis: Borel homomorphisms of second-countable locally compact groups are continuous

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group with left Haar measure $μ$ . (i) If $A \subseteq G$ is Borel with $0 < μ (A) < \infty$ , then $A A^{- 1}$ contains an open neighbourhood of the identity. (ii) If $T$ is a second-countable topological group and $φ : G \to T$ is a Borel-measurable group homomorphism, then $φ$ is continuous. In particular, a Borel homomorphism from a second-countable locally compact group into the unitary group $U (K)$ of a separable Hilbert space, with the strong operator topology, is strongly continuous.

Facts & Assumptions

Given: AC, a second-countable LCH group $G$ with left Haar measure $μ$ ; in part (ii) a second-countable topological group $T$ and a Borel homomorphism $φ$ .

[F1]

$G$ carries a nonzero left Haar measure $μ$ , positive on nonempty open sets and finite on compact sets; left translates of Borel sets preserve $μ$ (Existence of a left Haar integral, Left Haar integral and left Haar measure, Haar measure is positive on nonempty open sets and finite on compact sets).

[F2]

For $f \in L^{2} (G)$ the map $g \mapsto L_{g} f$ , $L_{g} f (x) = f (g^{- 1} x)$ , is norm continuous; hence so is $g \mapsto ⟨ L_{g} f, f ⟩ = \int_{G} f (g^{- 1} x) \overline{f (x)} d x$ (Strong continuity of left and modular right translations on L1 and L2).

[F3]

In a topological group, inversion is continuous, multiplication is continuous, every neighbourhood of the identity contains a symmetric neighbourhood, and a homomorphism continuous at the identity is continuous everywhere (Topological group: multiplication and inversion are continuous).

[F4]

The Borel $σ$ -algebra is generated by the open sets, and a Borel homomorphism is a group homomorphism measurable for the Borel structures; preimages of open sets are Borel (The Borel sigma-algebra of a topological space, A measurable function between measurable spaces).

[F5]

A second-countable space is Lindelöf: for an open cover, the members of a fixed countable base contained in some member of the cover form a countable refinement, and choosing one containing cover member for each of them uses Countable Choice, which follows from AC (Second countability: an at most countable basis for the topology, Compact implies countably compact, Lindel"of and limit point compact; countably compact together with Lindel"of implies compact; and, at the cost of countable or dependent choice, sequentially compact implies countably compact, countably compact implies limit point compact, and the converse holds when every singleton is closed, AC implies DC implies countable choice, The Axiom of Choice).

[F6]

The strong operator topology on $B (K)$ is the initial topology of the maps $T \mapsto T x$ , $x \in K$ (Strong and weak operator topologies); $U (K)$ is the group of unitary operators on $K$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F7]

A separable Hilbert space has a finite or countable orthonormal basis, which may be padded by zero vectors to a sequence indexed by $N$ (A Hilbert space with a dense sequence has a finite or countable orthonormal basis).

[F8]

$G$ is locally compact Hausdorff, so points have compact neighbourhoods and the open sets with compact closure form a base (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space).

Proof

technique · direct

Given: AC, the group $G$ with Haar measure $μ$ , and in part (ii) the group $T$ and the Borel homomorphism $φ$ .

1.1F1F2

Part (i): let $A$ be Borel with $0 < μ (A) < \infty$ and put $f = 1_{A} \in L^{2} (G)$ . For $g \in G$ , $L_{g} f = 1_{g A}$ , so $⟨ L_{g} f, f ⟩ = μ (g A \cap A)$ ; by [F2] this function of $g$ is continuous and equals $μ (A) > 0$ at $g = e$ . Hence there is an open neighbourhood $W$ of $e$ with $μ (g A \cap A) > 0$ , hence $g A \cap A \neq \emptyset$ , for every $g \in W$ ; writing $g a_{1} = a_{2}$ with $a_{1}, a_{2} \in A$ gives $g = a_{2} a_{1}^{- 1} \in A A^{- 1}$ . Thus $W \subseteq A A^{- 1}$ .

1.2F3

Part (ii), reduction: $φ$ is a homomorphism, so for $g_{0} \in G$ , $φ (g) φ (g_{0})^{- 1} = φ (g g_{0}^{- 1})$ ; since multiplication and inversion in $T$ are continuous by [F3], $φ$ is continuous at every point as soon as it is continuous at $e$ . It therefore suffices to show that $φ^{- 1} (U)$ is a neighbourhood of $e_{G}$ for every neighbourhood $U$ of $e_{T}$ .

1.3F5

A second-countable space is Lindelöf, with the argument of [F5]: fixing a countable base, the basic open sets contained in some member of an open cover form a countable refinement, and Countable Choice (a consequence of AC) selects a cover member for each of them. We apply this to $φ (G)$ with the subspace topology, which is second countable as a subspace of $T$ .

1.4F6F7

The unitary group $U (K)$ of a separable Hilbert space is second countable in the strong operator topology: fix a finite or countable orthonormal basis $(e_{n})$ padded to a sequence by [F7] and consider $Ψ : U (K) \to K^{N}$ , $Ψ (u) = (u e_{n})_{n}$ . It is injective (a unitary vanishing on a complete orthonormal system is zero) and continuous for the SOT by [F6]; conversely, if $u_{i} \to u$ in the initial topology of the coordinate maps $u \mapsto u e_{n}$ , then for $ξ = \sum_{n} c_{n} e_{n} \in K$ and $ε > 0$ choose $N$ with $\sum_{n > N} ∣ c_{n} ∣^{2} < ε^{2} / 16$ ; then $∥ (u_{i} - u) ξ ∥ \leq \sum_{n \leq N} ∣ c_{n} ∣ ∥ (u_{i} - u) e_{n} ∥ + 2 (\sum_{n > N} ∣ c_{n} ∣^{2})^{1 / 2} < ε / 2 + ε / 2$ for all $i$ beyond a suitable index, so $u_{i} \to u$ strongly. Thus the SOT on $U (K)$ is the initial topology of the countable family $(u \mapsto u e_{n})_{n}$ , making it homeomorphic to a subspace of the second-countable space $K^{N}$ , hence second countable.

1.5F6algebra

$U (K)$ with the strong operator topology is a topological group: if $u_{i} \to u$ and $v_{i} \to v$ strongly, then $∥ (u_{i} v_{i} - u v) ξ ∥ \leq ∥ u_{i} (v_{i} - v) ξ ∥ + ∥ (u_{i} - u) v ξ ∥ \leq ∥ (v_{i} - v) ξ ∥ + ∥ (u_{i} - u) v ξ ∥ \to 0$ ; and if $u_{i} \to u$ strongly with $u_{i}, u$ unitary, then $∥ (u_{i}^{- 1} - u^{- 1}) ξ ∥ = ∥ u_{i}^{- 1} (u - u_{i}) u^{- 1} ξ ∥ = ∥ (u - u_{i}) u^{- 1} ξ ∥ \to 0$ , so inversion is continuous.

2.1step 1.3F3

Fix a neighbourhood $U$ of $e_{T}$ and choose a symmetric neighbourhood $V$ of $e_{T}$ with $V^{2} \subseteq U$ , using continuity of multiplication at $(e_{T}, e_{T})$ and symmetry of neighbourhoods ([F3]). The family ${φ (g) V \cap φ (G) : g \in G}$ is an open cover of $φ (G)$ , because $φ (g) \in φ (g) V$ ; by [step 1.3] it has a countable subcover with centres $φ (g_{n})$ , $n \in N$ , chosen with $g_{n} \in G$ .

3.1step 2.1F1F4

The preimage $φ^{- 1} (V)$ has positive measure: it is Borel by [F4], and if $μ (φ^{- 1} (V)) = 0$ , then for every $g \in G$ the left translate $g φ^{- 1} (V)$ is null by [F1] and the sets $g_{n} φ^{- 1} (V)$ cover $G$ , since $φ (g) \in φ (g_{n}) V$ gives $φ (g_{n})^{- 1} φ (g) = φ (g_{n}^{- 1} g) \in V$ , that is, $g_{n}^{- 1} g \in φ^{- 1} (V)$ and $g \in g_{n} φ^{- 1} (V)$ . A countable cover of the nonempty open set $G$ by null sets would give $μ (G) = 0$ , contradicting positivity of $μ$ on the nonempty open set $G$ in [F1].

4.1step 3.1F1F8

Choose a Borel set $A \subseteq φ^{- 1} (V)$ with $0 < μ (A) < \infty$ : by [F8] and second countability, the members of a countable base with compact closure cover $G$ , so $G = ⋃_{m} K_{m}$ with $K_{m}$ compact; if $μ (φ^{- 1} (V) \cap K_{m}) = 0$ for every $m$ then countable additivity would give $μ (φ^{- 1} (V)) = 0$ , contrary to [step 3.1], so some $A : = φ^{- 1} (V) \cap K_{m}$ is Borel with $0 < μ (A) \leq μ (K_{m}) < \infty$ by [F1].

5.1step 1.1step 1.2step 4.1algebra

By part (i), [step 1.1], the set $A A^{- 1}$ contains an open neighbourhood of $e_{G}$ , and $A A^{- 1} \subseteq φ^{- 1} (V) φ^{- 1} (V)^{- 1} \subseteq φ^{- 1} (V V^{- 1}) = φ^{- 1} (V^{2}) \subseteq φ^{- 1} (U)$ , because $φ$ is a homomorphism and $V$ is symmetric. Hence $φ^{- 1} (U)$ is a neighbourhood of $e_{G}$ ; by [step 1.2] $φ$ is continuous. This proves (ii).

6.1step 1.1step 5.1step 1.4step 1.5∎

Let $φ : G \to U (K)$ be a Borel homomorphism from the second-countable LCH group $G$ into $U (K)$ with the strong operator topology. By [step 1.4] $U (K)$ is second countable and by [step 1.5] it is a topological group, so part (ii) proved in [step 5.1] applies and $φ$ is strongly continuous. Together with part (i) of [step 1.1] this proves every assertion of the statement.

Remarks

The proof of (ii) uses only the positive measure of the preimage of a neighbourhood of the identity, extracted through a countable subcover of the orbit cover; no countability of the group of values is assumed beyond second countability of the target.

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Unitary intertwiners preserve fibre multiplicity over a standard Borel base

Statement

Assume AC. Let $X$ be a standard Borel space, $μ$ a nonzero finite Borel measure on $X$ , and let $m, m^{'} : X \to {1, 2, \dots} \cup {\infty}$ be Borel multiplicity functions. If there is a unitary $U : L^{2} (X, μ; m) ⟶ L^{2} (X, μ; m^{'})$ with $U M_{f} = M_{f} U$ for every bounded Borel $f : X \to C$ , then $m = m^{'}$ $μ$ -almost everywhere. Consequently a multiplicity model of a projection-valued measure over a fixed base is unique in multiplicity, and a unitary intertwiner of two such models is a decomposable operator whose fibres are unitary almost everywhere.

Facts & Assumptions

Given: AC, a standard Borel space $X$ with a nonzero finite Borel measure $μ$ , Borel multiplicity functions $m, m^{'}$ , and a unitary $U : L^{2} (X, μ; m) \to L^{2} (X, μ; m^{'})$ with $U M_{f} = M_{f} U$ for all bounded Borel $f$ .

[F1]

For a Borel function $k : X \to N \cup {\infty}$ the field with fibre $C^{k (x)}$ and fundamental family $e_{j} (x) =$ the $j$ -th coordinate vector for $j \leq k (x)$ and $0$ otherwise has Borel Gram coefficients $x \mapsto δ_{i j} 1_{j \leq k (x)}$ and spans a dense subspace of each fibre; its direct integral $L^{2} (X, μ; k)$ is a Hilbert space of measurable square-integrable sections, and in the constant case $k \equiv r$ the fibre family $e_{1}, \dots, e_{r}$ is orthonormal and complete, so Parseval in each fibre makes $[ξ] \mapsto (⟨ ξ, e_{j} ⟩)_{j \leq r}$ an isometry onto the vector-valued $L^{2}$ -space $⨁_{j \leq r} L^{2} (X, μ)$ (Measurable Hilbert field from a countable fundamental family, Direct integral of a measurable Hilbert field, Measurable sections have measurable pointwise inner products, Parseval equivalences for an orthonormal family).

[F2]

For $f \in L^{\infty} (X, μ)$ the multiplication $M_{f}$ is a bounded operator on $L^{2} (X, μ; k)$ , $∥ M_{f} ∥ \leq ∥ f ∥_{\infty}$ , and for a Borel $B$ the operator $M_{1_{B}}$ is the orthogonal projection onto the closed subspace of sections supported in $B$ (Direct integral of a measurable Hilbert field, Hilbert space).

[F3]

On a $σ$ -finite standard Borel base with a measurable Hilbert field, the commutant of the diagonal multiplications $D = {M_{f} : f \in L^{\infty}}$ is exactly the set of decomposable operators; an operator commuting with $D$ is induced by a weakly measurable, essentially bounded field of fibre operators, and that field is unique up to a null set (Decomposable operators are the commutant of diagonal multiplication, Measurable and decomposable operator fields, Measurable essentially bounded operator fields act decomposably).

[F4]

A unitary between complex inner product spaces is a bijective linear isometry, so it exists only between fibres of equal dimension $k = k^{'}$ in $N \cup {\infty}$ : a finite-dimensional $C^{k}$ cannot be linearly isomorphic to $C^{\infty}$ , and $C^{k} ≇ C^{k^{'}}$ for distinct finite $k, k^{'}$ (Hilbert space, Separability: the existence of an at most countable dense subset, Orthonormal families, complete orthonormal systems and Hilbert bases).

[F5]

The sets ${x : k (x) > j}$ are Borel for a Borel $k$ , and $X$ is the countable disjoint union of the Borel sets $B_{k, k^{'}} = {m = k} \cap {m^{'} = k^{'}}$ , so measures on $X$ are countably additive over this partition; the standard Borel base is $σ$ -finite for the finite measure $μ$ (Standard Borel spaces, Monotone convergence for the integral, A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density).

Proof

technique · direct

Given: AC, the data and the unitary $U$ of the statement.

1.1F2

For each Borel set $B \subseteq X$ , the identity $U M_{1_{B}} = M_{1_{B}}^{'} U$ and unitarity of $U$ give $M_{1_{B}}^{'} = U M_{1_{B}} U^{- 1}$ , so $U$ carries the range of the projection $M_{1_{B}}$ onto the range of $M_{1_{B}}^{'}$ ; restricting to these closed subspaces yields a unitary $U_{B}$ from the sections of $L^{2} (X, μ; m)$ supported in $B$ to the sections of $L^{2} (X, μ; m^{'})$ supported in $B$ .

2.1step 1.1F1F2

Fix $k, k^{'} \in N \cup {\infty}$ and put $B = B_{k, k^{'}}$ , a Borel set by [F5]. The supported subspace of $L^{2} (X, μ; m)$ over $B$ is, by [F1], the direct integral over $(B, μ ∣_{B})$ of the constant field with fibre $C^{k}$ ; similarly the target over $B$ is the constant field with fibre $C^{k^{'}}$ . The unitary $U_{B}$ of [step 1.1] intertwines the multiplication operators for all bounded Borel $f$ on $B$ , because $U_{B}$ is the restriction of $U$ and $M_{f}$ preserves the supported subspaces.

3.1step 2.1F3

Assume $μ (B) > 0$ . Apply [F3] to the sum field $C^{k} \oplus C^{k'}$ and its block operator whose only nonzero block is $U_{B}$ from the first summand to the second. This operator commutes with all diagonal multiplications, so its off-diagonal block is decomposable: there is a weakly measurable, essentially bounded operator field $x \mapsto T_{x} \in B (C^{k}, C^{k^{'}})$ with $U_{B}$ acting by $T_{x}$ fibrewise; the same applies to $U_{B}^{*} = U_{B}^{- 1}$ , and since $U_{B}^{*} U_{B} = I$ and $U_{B} U_{B}^{*} = I$ , the uniqueness of decomposable fields in [F3] gives $T_{x}^{*} T_{x} = I_{k}$ and $T_{x} T_{x}^{*} = I_{k^{'}}$ for $μ$ -almost every $x \in B$ . Thus for almost every $x$ the fibre map $T_{x}$ is a unitary between $C^{k}$ and $C^{k^{'}}$ .

4.1step 3.1F4

Hence $k = k^{'}$ whenever $μ (B_{k, k^{'}}) > 0$ : by [step 3.1] a unitary $C^{k} \to C^{k^{'}}$ exists for some $x$ , and [F4] says this forces $k = k^{'}$ .

5.1step 4.1F5

Therefore ${m \neq m^{'}} = ⋃_{k \neq k^{'}} B_{k, k^{'}}$ is a countable union of sets of $μ$ -measure zero, hence $μ$ -null by countable additivity; that is, $m = m^{'}$ $μ$ -almost everywhere.

6.1step 5.1F3∎

The intertwiner $U$ itself is decomposable: its block operator on the direct sum of the two fields commutes with all diagonal multiplications, so [F3] represents its off-diagonal block by a weakly measurable essentially bounded field. Applying the same to $U^{- 1} = U^{*}$ , whose field is the fibrewise adjoint up to a null set by the uniqueness clause of [F3], and using $U^{*} U = U U^{*} = I$ as in [step 3.1], its fibres are unitary almost everywhere. Thus every unitary intertwiner of two multiplicity models over the fixed base $(X, μ)$ has unitary fibres a.e., and the multiplicity is unique, which is exactly the rigidity statement a multiplicity model of a projection-valued measure over a fixed base invokes.

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A system of imprimitivity integrates to a nondegenerate representation of the transformation algebra

Statement

Assume AC. Let $(U, P)$ be a system of imprimitivity on a second-countable locally compact Hausdorff $G$ -space $X$ with continuous action, as required by the transformation-algebra definition. For $f \in C_{c} (G \times X)$ define $π (f)$ by the scalar pairing $⟨ π (f) ξ, η ⟩ = \int_{G} \int_{X} f (g, x) d E_{U_{g} ξ, η} (x) d g (ξ, η \in H),$ where $E_{ξ, η} (B) = ⟨ P (B) ξ, η ⟩$ . Then $π (f)$ is a bounded operator, $∥ π (f) ∥ \leq \int_{G} ∥ f (g, \cdot) ∥_{\infty} d g$ , the map $f \mapsto π (f)$ is a $*$ -representation of the transformation algebra $C_{c} (G \times X)$ , and it is nondegenerate: the closed span of $π (C_{c} (G \times X)) H$ is $H$ .

Facts & Assumptions

Given: AC, the system of imprimitivity $(U, P)$ on the second-countable LCH $G$ -space $X$ with continuous action, and $f, f_{1}, f_{2}, f_{3} \in C_{c} (G \times X)$ .

[F1]

For a bounded Borel $b : X \to C$ and $ξ, η \in H$ the operator $M_{b} = \int b d P$ satisfies $⟨ M_{b} ξ, η ⟩ = \int b d E_{ξ, η}$ , $M_{b_{1} b_{2}} = M_{b_{1}} M_{b_{2}}$ , $M_{\overset{ˉ}{b}} = M_{b}^{*}$ , $∥ M_{b} ∥ \leq ∥ b ∥_{\infty}$ and $M_{1} = I$ ; $E_{ξ, η}$ is a finite complex measure with $∣ E_{ξ, η} ∣ (X) \leq ∥ ξ ∥ ∥ η ∥$ ; if bounded Borel $b_{n} \to b$ pointwise $P$ -a.e. and $\sup_{n} ∥ b_{n} ∥_{\infty} < \infty$ then $M_{b_{n}} \to M_{b}$ strongly (Bounded borel pvm integral, Scalar and complex measures from a pvm, Pvm integral is a star homomorphism).

[F2]

$(U, P)$ is a strongly continuous unitary representation together with a PVM satisfying $U_{g} P (E) U_{g}^{- 1} = P (g E)$ for all $g$ and Borel $E$ ; equivalently $U_{g} M_{b} U_{g}^{- 1} = M_{b \circ g^{- 1}}$ for every bounded Borel $b$ , i.e. $U_{g} M_{b} = M_{b \circ g^{- 1}} U_{g}$ (Systems of imprimitivity for a Borel $G$ -space).

[F3]

The transformation algebra has product $(f_{1} * f_{2}) (g, x) = \int_{G} f_{1} (h, x) f_{2} (h^{- 1} g, h^{- 1} x) d h$ and involution $f^{*} (g, x) = Δ_{G} (g)^{- 1} \overline{f (g^{- 1}, g^{- 1} x)}$ (The transformation (covariance) algebra $C_{c} (G \times X)$ , Modular function of a locally compact group, Compactly supported convolution on a group).

[F4]

The Haar integral is left invariant and finite on compacta, and $\int_{G} w (x^{- 1}) d x = \int_{G} Δ_{G} (x^{- 1}) w (x) d x$ for nonnegative Borel $w$ (Left Haar integral and left Haar measure, Haar change of variables under inversion, Haar measure is positive on nonempty open sets and finite on compact sets).

[F5]

A continuous function with compact support is uniformly continuous on compacta: for $K_{G}, K_{X}$ compact there is for each $ε > 0$ a neighbourhood of every $(g_{0}, x_{0})$ on which $∣ f - f (g_{0}, x_{0}) ∣ < ε$ ; consequently $g \mapsto f (g, \cdot)$ is continuous in the supremum norm on a neighbourhood of each $g_{0}$ , with supports in a fixed compact subset of $X$ and vanishing outside the compact projection of $supp f$ (Compact support, $C_{c} (X)$ , and $C_{0} (X)$ ).

[F6]

Bochner calculus in the Hilbert space $H$ : a strongly measurable $H$ -valued function with finite integral of the norm is Bochner integrable, $∥ \int F ∥ \leq \int ∥ F ∥$ , and bounded linear maps commute with $\int$ (Bochner integrability criterion, Bochner integral norm inequality, Bounded linear maps commute with Bochner integration). Scalar iterated integrals of bounded integrable kernels agree (Fubini's theorem for L^1 functions on a sigma-finite product).

[F7]

There is a contractively bounded approximate identity $e_{U} \in C_{c} (G)$ , $e_{U} \geq 0$ , $supp e_{U} \subseteq U$ , $∥ e_{U} ∥_{1} = 1$ , directed by identity neighbourhoods of $G$ , with $e_{U} * f \to f$ in $L^{1}$ (L1 group algebras have a contractively bounded approximate identity).

[F8]

$X$ is second-countable LCH, so it is the union of an increasing sequence of compact sets $K_{n}$ (replace a countable compact cover by its successive finite unions) and for each $n$ there is $b_{n} \in C_{c} (X)$ with $0 \leq b_{n} \leq 1$ and $b_{n} = 1$ on $K_{n}$ (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel for the Polish/compact-exhaustion structure, LCH Urysohn cutoff).

Proof

technique · direct

Given: AC, the system $(U, P)$ and a test function $f \in C_{c} (G \times X)$ .

1.1F1F2F5

For fixed $g$ put $M_{f (g, \cdot)} = \int_{X} f (g, \cdot) d P$ ; by [F1] this is a bounded operator and $∥ M_{f (g, \cdot)} ∥ \leq ∥ f (g, \cdot) ∥_{\infty}$ . The map $g \mapsto M_{f (g, \cdot)}$ is norm continuous: if $K_{G}$ is the compact group projection of $supp f$ , then for $g, g_{0} \in K_{G}$ and $ε > 0$ uniform continuity of $f$ on the compact set $K_{G} \times (compact x -support)$ gives $∣ f (g, x) - f (g_{0}, x) ∣ \leq ε$ for all $x$ once $g$ is close to $g_{0}$ , whence $∥ M_{f (g, \cdot)} - M_{f (g_{0}, \cdot)} ∥ \leq ∥ f (g, \cdot) - f (g_{0}, \cdot) ∥_{\infty} \leq ε$ ; and $M_{f (g, \cdot)} = 0$ for $g \notin K_{G}$ . Consequently $g \mapsto M_{f (g, \cdot)} U_{g} ξ$ is strongly continuous for every $ξ \in H$ (product of a norm-continuous and a strongly continuous factor) and supported in the compact set $K_{G}$ .

1.2F2

Covariance in operator form: conjugating $M_{b} = \int b d P$ by the unitary $U_{h}$ and using $U_{h} P (E) U_{h}^{- 1} = P (h E)$ gives $U_{h} M_{b} U_{h}^{- 1} = \int b (h^{- 1} x) d P (x) = M_{b \circ h^{- 1}}$ , that is $U_{h} M_{b} = M_{b \circ h^{- 1}} U_{h}$ for every bounded Borel $b$ and $h \in G$ .

1.3F6F7

Nondegeneracy, first factor: for every $ξ \in H$ , $∥ U (e_{U}) ξ - ξ ∥ \to 0$ along the approximate identity of [F7], where $U (a) : = \int_{G} a (g) U_{g} d g$ is the Bochner integral in $H$ ; indeed $∥ U (e_{U}) ξ - ξ ∥ \leq \int_{G} e_{U} (g) ∥ U_{g} ξ - ξ ∥ d g$ and, given $ε > 0$ , strong continuity gives an identity neighbourhood $V$ with $∥ U_{g} ξ - ξ ∥ < ε$ for $g \in V$ , while for $U \subseteq V$ the support condition and $∥ e_{U} ∥_{1} = 1$ make the last integral at most $ε$ .

2.1F1F6step 1.1

Define $π (f) ξ : = \int_{G} M_{f (g, \cdot)} U_{g} ξ d g$ as a Bochner integral: strong measurability follows from [step 1.1], and $\int_{G} ∥ M_{f (g, \cdot)} U_{g} ξ ∥ d g \leq ∥ f ∥_{1, \infty} ∥ ξ ∥$ with $C (f) : = \int_{G} ∥ f (g, \cdot) ∥_{\infty} d g < \infty$ because the integrand vanishes off $K_{G}$ and is bounded there by $∥ f ∥_{\infty}$ on a compact set of finite Haar measure. Hence $π (f)$ is a well-defined bounded operator with $∥ π (f) ξ ∥ \leq C (f) ∥ ξ ∥$ for every $ξ$ , so $∥ π (f) ∥ \leq C (f)$ . Pairing with $η$ and commuting the bounded functional $⟨ \cdot, η ⟩$ through the Bochner integral gives exactly the displayed identity $⟨ π (f) ξ, η ⟩ = \int_{G} ⟨ M_{f (g, \cdot)} U_{g} ξ, η ⟩ d g = \int_{G} \int_{X} f (g, x) d E_{U_{g} ξ, η} (x) d g$ . The map $f \mapsto π (f)$ is complex-linear because the integrand is bilinear in $(f, ξ)$ and the Bochner integral is linear.

2.2F1F6step 1.3

Nondegeneracy, second factor: $M_{b_{n}} \to I$ strongly for the sequence of [F8], by the pointwise dominated convergence of [F1], since $b_{n} \to 1$ pointwise on $X$ and $0 \leq b_{n} \leq 1$ . For the product function $(g, x) \mapsto e_{U} (g) b_{n} (x)$ one has $π (e_{U} \otimes b_{n}) = M_{b_{n}} U (e_{U})$ , because the bounded operator $M_{b_{n}}$ commutes with the Bochner integral $\int_{G} e_{U} (g) M_{b_{n}} U_{g} d g = M_{b_{n}} \int_{G} e_{U} (g) U_{g} d g$ . Given $ξ$ and $ε > 0$ choose $n$ with $∥ M_{b_{n}} ξ - ξ ∥ < ε / 2$ and then $U$ small enough that $∥ U (e_{U}) ξ - ξ ∥ < ε / 2$ ; then $∥ π (e_{U} \otimes b_{n}) ξ - ξ ∥ < ε$ . Hence the closed span of $π (C_{c} (G \times X)) H$ contains every $ξ$ , so $π$ is nondegenerate.

3.1F3F6step 2.1step 1.2

Multiplicativity: for $ξ, η \in H$ , using [step 2.1] twice, [step 1.2] with $b = f_{2} (r, \cdot)$ and $h$ , and the left-Haar substitution $r = h^{- 1} g$ (so $h r = g$ , $d r = d g$ ) one computes $⟨ π (f_{1}) π (f_{2}) ξ, η ⟩ = \int_{G} \int_{G} ⟨ M_{f_{1} (h, \cdot)} M_{f_{2} (r, h^{- 1} \cdot)} U_{h r} ξ, η ⟩ d r d h = \int_{G} \int_{G} ⟨ M_{f_{1} (h, \cdot) f_{2} (h^{- 1} g, h^{- 1} \cdot)} U_{g} ξ, η ⟩ d g d h$ ; the scalar kernel is integrable on the compact support, so Fubini's theorem turns the iterated integral into $\int_{G} ⟨ M_{\int_{G} f_{1} (h, \cdot) f_{2} (h^{- 1} g, h^{- 1} \cdot) d h} U_{g} ξ, η ⟩ d g = ⟨ π (f_{1} * f_{2}) ξ, η ⟩$ , using the identification of the inner Bochner integral of multiplication operators through its pairings and the definition of the twisted product in [F3]. As $η$ is arbitrary this gives $π (f_{1}) π (f_{2}) = π (f_{1} * f_{2})$ .

3.2F1F3F4step 2.1step 1.2

Adjoint: taking adjoints in the defining Bochner integral and using $U_{g}^{*} = U_{g^{- 1}}$ and $M_{b}^{*} = M_{\overset{ˉ}{b}}$ , $π (f)^{*} = \int_{G} U_{g}^{*} M_{\overline{f (g, \cdot)}} d g = \int_{G} M_{\overline{f (g, g \cdot)}} U_{g^{- 1}} d g$ , where the second equality is [step 1.2] with $b = \overline{f (g, \cdot)}$ and $h = g^{- 1}$ . Substituting $h = g^{- 1}$ in the Haar integral and using [F4] in the form $\int_{G} w (g) d g = \int_{G} Δ_{G} (h)^{- 1} w (h^{- 1}) d h$ gives $π (f)^{*} = \int_{G} M_{Δ_{G} (h)^{- 1} \overline{f (h^{- 1}, h^{- 1} \cdot)}} U_{h} d h = π (f^{*})$ , with $f^{*}$ the modular involution of [F3].

4.1step 2.1step 3.1step 3.2step 2.2F8∎

Steps 2.1, 3.1 and 3.2 show that $f \mapsto π (f)$ is a bounded $*$ -representation of the transformation algebra with the stated norm bound, and step 2.2 shows it is nondegenerate. The homogeneous-space case $X = G / H$ of the pair satisfies the added topological hypotheses, since $G / H$ is second-countable LCH with continuous left action (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel).

Remarks

The pairing definition and the operator definition agree, and no regularity of $P$ beyond the PVM axioms is used; the continuous action is needed only to make $f (g, \cdot)$ vary continuously in the supremum norm.

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Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ a closed subgroup, and $q : G \to G / H$ the quotient map. Then there is a Borel cross-section $s : G / H \to G$ with $q \circ s = {i d}_{G / H}$ , and for every $g \in G$ there is a unique $h (g, x) \in H$ with $g s (x) = s (g x) h (g, x)$ , where $x \in G / H$ ; the map $(g, x) \mapsto h (g, x)$ is Borel into $H$ . Consequently the map $G \to G / H \times H$ , $g \mapsto (q (g), s (q (g))^{- 1} g)$ , is a Borel isomorphism onto $G / H \times H$ , and the quasi-invariant measure class on $G / H$ may be transported to a $σ$ -finite measure on $G$ along $s$ .

Facts & Assumptions

Given: AC, a second-countable LCH group $G$ , a closed subgroup $H \leq G$ , and the quotient map $q : G \to G / H$ .

[F1]

$G$ and $G / H$ are Polish, $q$ is continuous and open, and $G / H$ is locally compact Hausdorff (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Compact lifts and averaging onto C_c(G/H)).

[F2]

A Polish space carries a compatible complete metric $d$ ; convergent sequences, Cauchy sequences, closed sets, diameters and compactness are read in this metric (Polish spaces are separable completely metrizable spaces, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, Complete metric space: every Cauchy sequence converges in the space).

[F3]

$G$ has a countable base of relatively compact open sets, and for every point $g$ and open $U ∋ g$ there is a base element $V$ with $g \in V \subseteq \overline{V} \subseteq U$ and compact closure, with $diam V$ as small as prescribed (Second countability: an at most countable basis for the topology, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Countably compact, Lindel"of, sequentially compact, limit point compact and $σ$ -compact spaces, and relatively compact subsets).

[F4]

By [F1], $G$ and $G / H$ are Polish and hence standard Borel; their Borel structures are generated by their open sets (Standard Borel spaces, The Borel sigma-algebra of a topological space).

[F5]

The quotient map satisfies $q (g s) = g q (s)$ for all $g, s \in G$ , and $H = {g : q (g) = e H}$ is the stabilizer of the identity coset; multiplication and inversion of $G$ are continuous, so composites of Borel maps with the group operations are Borel (Left and right cosets $g H$ and $H g$ of a subgroup, Topological group: multiplication and inversion are continuous, The Borel sigma-algebra of a topological space).

[F6]

Countable Choice, a consequence of the standing AC, selects one point from each member of a countable family of nonempty sets, and in particular fixes a point in each member of a countable base of $G$ (The Axiom of Choice, Compact implies countably compact, Lindel"of and limit point compact; countably compact together with Lindel"of implies compact; and, at the cost of countable or dependent choice, sequentially compact implies countably compact, countably compact implies limit point compact, and the converse holds when every singleton is closed).

Proof

technique · construct

Given: AC, the second-countable LCH group $G$ , the closed subgroup $H$ , and the quotient map $q$ .

1.1F2F3F6

Fix a compatible complete metric $d$ on $G$ by [F2]. By second countability and [F3], choose a countable base $(V_{k})_{k \in N}$ of relatively compact open sets such that for every open $U \subseteq G$ , every $g \in U$ , and every $ϵ > 0$ there is a $V_{k}$ with $g \in V_{k} \subseteq \overline{V_{k}} \subseteq U$ and $diam V_{k} < ϵ$ : for each basic open set and each positive integer $m$ , the refinements supplied by [F3] cover that basic open set, and second countability gives a countable subcover. By [F6] choose a point $c_{k} \in V_{k}$ for every $k$ .

1.2F3F1F5

For $x \in G / H$ define indices $k_{n} (x)$ by recursion: $k_{0} (x)$ is the least $k$ with $x \in q (V_{k})$ and $diam V_{k} \leq 1$ ; such an index exists by choosing any $g \in q^{- 1} (x)$ and applying [F3] inside $G$ . Having defined $k_{n} (x)$ , let $k_{n + 1} (x)$ be the least $k$ with $\overline{V_{k}} \subseteq V_{k_{n} (x)}$ , $x \in q (V_{k})$ , and $diam V_{k} \leq 2^{- (n + 1)}$ . This index exists by choosing $g \in q^{- 1} (x) \cap V_{k_{n} (x)}$ and applying [F3] inside $V_{k_{n} (x)}$ with the prescribed diameter. Each $k_{n}$ is Borel: for $n = 0$ each candidate condition on $x$ is membership in the open set $q (V_{k})$ , and inductively the set of $x$ with $\overline{V_{k}} \subseteq V_{k_{n} (x)}$ is the union over countably many $j$ with $\overline{V_{k}} \subseteq V_{j}$ of ${x : k_{n} (x) = j}$ ; intersecting with $q (V_{k})$ and the fixed diameter condition gives Borel candidate sets. Selecting the least index in a Borel family is Borel.

2.1F2step 1.2

The points $p_{n} (x) : = c_{k_{n} (x)}$ are Borel functions of $x$ and converge: for $m \geq n$ , $p_{m} (x) \in V_{k_{m} (x)} \subseteq \overline{V_{k_{n} (x)}}$ , so $d (p_{m} (x), p_{n} (x)) \leq diam \overline{V_{k_{n} (x)}} \leq 2^{- n}$ . Completeness gives a limit $s (x) \in ⋂_{n} \overline{V_{k_{n} (x)}}$ , whose diameter is zero. The map $s$ is Borel: for each nonempty closed $F \subseteq G$ , $s (x) \in F$ iff $d (p_{n} (x), F) \to 0$ ; this condition is a countable intersection of countable unions of Borel sets because $p_{n}$ is Borel and distance to $F$ is continuous. The empty closed set has empty preimage.

3.1step 1.2F1F2F6step 2.1

$s$ is a cross-section: for each $n$ choose $g_{n} \in V_{k_{n} (x)}$ with $q (g_{n}) = x$ , which is possible because $x \in q (V_{k_{n} (x)})$ by [step 1.2]. Both $g_{n}$ and $s (x)$ lie in $\overline{V_{k_{n} (x)}}$ , a set of diameter at most $2^{- n}$ , so $d (g_{n}, s (x)) \leq 2^{- n} \to 0$ and the sequence $g_{n}$ converges to $s (x)$ ; continuity of $q$ gives $x = q (g_{n}) \to q (s (x))$ , and since the left-hand sequence is constant, $q (s (x)) = x$ .

4.1F1F4step 2.1step 3.1

Transport of measures: for a $σ$ -finite Borel measure $μ$ on $G / H$ , define $ν (E) : = μ (s^{- 1} (E))$ for Borel $E \subseteq G$ . This is a Borel measure because $s$ is Borel. It is carried by the Borel image $s (G / H) = {g \in G : s (q (g)) = g}$ : the latter is Borel since $g \mapsto (s (q (g)), g)$ is Borel and the diagonal of the Polish space $G$ is closed. If $G / H = ⋃_{n} A_{n}$ with $μ (A_{n}) < \infty$ , then $s (A_{n}) = s (G / H) \cap q^{- 1} (A_{n})$ is Borel and $ν (s (A_{n})) = μ (A_{n})$ , since $q \circ s = i d$ . The sets $s (A_{n})$ together with $G ∖ s (G / H)$ cover $G$ , and the last set has $ν$ -measure zero, so $ν$ is $σ$ -finite. Equivalent measures have equivalent pushforwards by the definition of $ν$ , so the measure class is transported along $s$ .

4.2step 3.1F5

For $g \in G$ and $x \in G / H$ one has $q (g s (x)) = g q (s (x)) = g x = q (s (g x))$ by [step 3.1], so $s (g x)^{- 1} g s (x) \in q^{- 1} (e H) = H$ . Thus $h (g, x) = s (g x)^{- 1} g s (x)$ is the unique element of $H$ with $g s (x) = s (g x) h (g, x)$ . The map $(g, x) \mapsto h (g, x)$ is Borel as a composite of the Borel map $(g, x) \mapsto s (g x)$ (composition of $s$ with the continuous action map) with the continuous group operations.

4.3F5step 3.1

The map $Φ : G \to G / H \times H$ , $g \mapsto (q (g), s (q (g))^{- 1} g)$ , is Borel and so is $Ψ : G / H \times H \to G$ , $(x, h) \mapsto s (x) h$ ; they are mutually inverse: $q (s (x) h) = q (s (x)) = x$ and $s (x)^{- 1} s (x) h = h$ , while $s (q (g)) (s (q (g))^{- 1} g) = g$ . Hence $Φ$ is a Borel isomorphism onto $G / H \times H$ .

5.1step 3.1step 4.2step 4.3step 4.1step 1.1∎

We have constructed a Borel cross-section $s$ of $q$ [step 3.1], the unique Borel section cocycle $h (g, x) = s (g x)^{- 1} g s (x)$ [step 4.2], the Borel isomorphism $G ≅ G / H \times H$ [step 4.3], and the transported $σ$ -finite measure [step 4.1]. AC is inherited through the Polish-space input [F1], whose complete-metrization proof uses DC and the ultrafilter lemma, and supplies Countable Choice for the countable-base refinements and base points $c_{k}$ in step 1.1 and the fibre points $g_{n}$ in step 3.1. The least-index recursion and the Borel cocycle and product formulas add no choice requirement.

Remarks

The section is constructed from a countable base by a deterministic least-index recursion, so its Borelness is proved rather than assumed; the compatibility of the presented metric with the group topology is the only metric input.

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LCA Fourier transforms form a dense algebra in C0 of the dual

Statement

Assume AC and let $N$ be a second-countable LCH abelian group. With $\hat{f} (χ) = \int_{N} f (n) χ (n) d n$ , each $f \in L^{1} (N)$ has $\hat{f} \in C_{0} (\hat{N})$ , and the functions $\hat{f}$ form a self-adjoint separating nowhere-vanishing algebra whose uniform closure is $C_{0} (\hat{N})$ . No injectivity or inversion claim is needed.

Facts & Assumptions

Given: AC and a second-countable LCH abelian group $N$ with Haar measure.

[F1]

$L^{1} (N)$ is a complex Banach $*$ -algebra with convolution $*$ and involution $f^{*} (n) = Δ_{N} (n)^{- 1} \overline{f (n^{- 1})}$ ; the classification result for its characters says that $λ \mapsto λ (f) = \int f χ d n$ is a homeomorphism from $\hat{N}$ with the compact-open topology onto the character space of $L^{1} (N)$ with the pointwise-evaluation topology, and distinct characters give distinct characters of $L^{1} (N)$ (L1 of a locally compact group is a Banach star-algebra, Involution on L1 of a locally compact group, Characters of the L1 algebra of an abelian group).

[F2]

The sum-norm unitization $\tilde{A} = C \oplus L^{1} (N)$ is a nonzero unital complex Banach algebra, and every character of $\tilde{A}$ is unital and norm-bounded by $1$ ; its unit ball is weak-star compact by Banach–Alaoglu, and the character space is closed in that unit ball, hence compact Hausdorff (Characters on a unital Banach algebra are continuous, Banach–Alaoglu, Character and maximal ideal space).

[F3]

On a compact Hausdorff space, a self-adjoint separating subalgebra of $C (K)$ containing the constants has uniform closure $C (K)$ (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[F4]

A continuous function on a locally compact Hausdorff space vanishing at infinity extends by zero at the point at infinity to a continuous function on the one-point compactification (The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ ).

[F5]

Product and conjugation of Fourier transforms follow from Fubini and the involution: for $f, g$ in the dense subspace $C_{c} (N)$ one has $\hat{f * g} = \hat{f} \hat{g}$ and $\hat{f^{*}} = \overline{\hat{f}}$ , and both sides are bounded bilinear in $(f, g)$ with $∥ \hat{f} ∥_{\infty} \leq ∥ f ∥_{1}$ , so the identities hold on all of $L^{1} (N)$ (Fubini's theorem for L^1 functions on a sigma-finite product, Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F6]

Nonnegative compactly supported cutoffs exist near any point, and Haar measure is positive on nonempty open sets, so there is $c \in C_{c} (N)$ with $c \geq 0$ and $Re \hat{c} (χ) > 0$ for a prescribed $χ$ (LCH Urysohn cutoff, Haar measure is positive on nonempty open sets and finite on compact sets, The Pontryagin dual with the compact-open topology).

[F7]

AC is the standing hypothesis (The Axiom of Choice).

Proof

technique · direct

Given: AC, the second-countable LCH abelian group $N$ , its dual $\hat{N}$ , and $f \in L^{1} (N)$ .

1.1F1F5

The Fourier transform is well defined and bounded: $∣ \hat{f} (χ) ∣ \leq \int_{N} ∣ f ∣ d n = ∥ f ∥_{1}$ , and $\hat{f} : \hat{N} \to C$ is continuous, since compact-open convergence $χ_{i} \to χ$ gives uniform convergence on a compact set carrying all but $ε$ of $∣ f ∣ d n$ after choosing a compactly supported $L^{1}$ -approximant of $f$ .

1.2F1F2F4

Let $\tilde{A} = C \oplus L^{1} (N)$ be the sum-norm unitization and $K = Δ (\tilde{A})$ its character space, a compact Hausdorff space by [F2]. Every $ψ \in K$ is unital; if $ψ$ does not vanish on $L^{1} (N)$ its restriction is a character of $L^{1} (N)$ , hence equal to $λ_{χ}$ for exactly one $χ \in \hat{N}$ by [F1], and then $ψ (z, f) = z + λ_{χ} (f)$ ; otherwise $ψ (1, 0) = 1$ and $ψ (0, f) = 0$ for all $f$ , so $ψ = q$ with $q (z, f) = z$ . Thus $K = {{\tilde{λ}}_{χ} : χ \in \hat{N}} \cup {q}$ , the map $χ \mapsto {\tilde{λ}}_{χ}$ is a homeomorphism onto the open subset $K ∖ {q}$ (openness because ${\tilde{λ}}_{χ} \mapsto {\tilde{λ}}_{χ} (0, f) = \hat{f} (χ)$ ), and $K$ is the one-point compactification of $\hat{N}$ in the sense of [F4].

2.1F2step 1.2

Each $\hat{f}$ lies in $C_{0} (\hat{N})$ : the evaluation function $ψ \mapsto ψ (0, f)$ is continuous on $K$ by its pointwise-evaluation topology, equals $\hat{f}$ on $\hat{N}$ , and is zero at $q$ . Hence its closed superlevel set ${ψ : ∣ ψ (0, f) ∣ \geq ε}$ is compact and misses $q$ , for every $ε > 0$ . This is a compact superlevel set of $\hat{f}$ in $\hat{N}$ , proving the required vanishing at infinity.

3.1F1F3F5F6step 1.2step 2.1

The algebra $A = {\hat{f} + c : f \in L^{1} (N), c \in C}$ of continuous functions on $K$ contains the constants, is self-adjoint and separates points: $\hat{f} \hat{g} + c$ corresponds to the $L^{1}$ -function $f * g$ up to constants, $\hat{f^{*}} = \overline{\hat{f}}$ by [F5], distinct points of $\hat{N}$ are separated by some $\hat{f}$ by [F1], and $q$ is separated from any ${\tilde{λ}}_{χ}$ by a $\hat{f}$ with $\hat{f} (χ) \neq 0$ , which exists by [F6]. By [F3] its uniform closure is $C (K)$ .

4.1F3step 3.1

The transforms alone have uniform closure $C_{0} (\hat{N})$ : if $g \in C_{0} (\hat{N})$ , regard it as an element of $C (K)$ with $g (q) = 0$ by [F4] and choose ${\hat{f}}_{n} + c_{n} \in A$ with $∥ {\hat{f}}_{n} + c_{n} - g ∥_{\infty} < ε$ ; evaluating at $q$ gives $∣ c_{n} ∣ < ε$ because ${\hat{f}}_{n} (q) = 0$ and $g (q) = 0$ , so $∥ {\hat{f}}_{n} - g ∥_{\infty} \leq ∥ {\hat{f}}_{n} + c_{n} - g ∥_{\infty} + ∣ c_{n} ∣ < 2 ε$ ; hence $g$ is a uniform limit of Fourier transforms.

5.1step 2.1step 3.1step 4.1F5F6F7∎

By [step 4.1] the Fourier transforms are uniformly dense in $C_{0} (\hat{N})$ ; by [step 2.1] each lies in $C_{0} (\hat{N})$ ; by [F5] the family is a self-adjoint algebra; it separates points by the separation used in [step 3.1]; and it vanishes nowhere by the bump construction of [F6], which at each $χ$ supplies $c$ with $\hat{c} (χ) \neq 0$ . No injectivity of the transform and no inversion formula were used.

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Multiplicity model of a projection-valued measure over a standard Borel base

Statement

Assume AC. Let $X$ be a standard Borel space, $P$ a projection-valued measure on $X$ acting on a nonzero separable complex Hilbert space $H$ , and let $μ$ be a finite Borel measure on $X$ that is $P$ -faithful, i.e. $P (E) = 0$ iff $μ (E) = 0$ . Then there exist a Borel function $m : X \to {1, 2, \dots} \cup {\infty}$ and a unitary $W : H ⟶ \int_{X}^{\oplus} C^{m (x)} d μ (x)$ such that $W P (E) W^{- 1} = M_{1_{E}}$ for every Borel $E \subseteq X$ . Such a $μ$ exists for every nonzero separable $H$ : for any dense sequence $(ξ_{j})$ with $ξ_{j} \neq 0$ , $μ = \sum_{j} 2^{- j} ∥ ξ_{j} ∥^{- 2} ⟨ P (\cdot) ξ_{j}, ξ_{j} ⟩$ is finite, $P$ -faithful and Borel. Any two $P$ -faithful measures are mutually absolutely continuous.

Facts & Assumptions

Given: AC, the standard Borel space $X$ , the PVM $P$ on the nonzero separable Hilbert space $H$ , and a finite $P$ -faithful Borel measure $μ$ on $X$ .

[F1]

For bounded Borel $b$ , $M_{b}^{P} = \int b d P$ satisfies $⟨ M_{b}^{P} ξ, η ⟩ = \int b d E_{ξ, η}$ and $∥ M_{b}^{P} ∥ \leq ∥ b ∥_{\infty}$ , with $M_{b c} = M_{b}^{P} M_{c}^{P}$ and $M_{\overset{ˉ}{b}} = (M_{b}^{P})^{*}$ ; each $E_{ξ, η}$ is a finite complex measure; projections $P (E) = M_{1_{E}}^{P}$ commute (Bounded borel pvm integral, Pvm integral is a star homomorphism, Scalar and complex measures from a pvm, Projection valued measure).

[F2]

Every standard Borel space $X$ admits a bimeasurable injection $c : X \to [0, 1]$ onto a Borel subset of $[0, 1]$ (Standard borel spaces admit bimeasurable real codings, Standard Borel spaces).

[F3]

The operator $S = \int_{X} c d P$ is bounded and self-adjoint, hence normal; it has a spectral PVM $E_{S}$ on the compact $σ (S) \subseteq [0, 1]$ with $\int z d E_{S} (z) = S$ and $E_{S} (D) = 1_{D} (S)$ for Borel $D$ given by the bounded Borel functional calculus. If $Q$ is a regular PVM on a nonempty compact $Λ \subseteq R$ with $\int z d Q (z) = S$ , then $Q (Λ ∖ σ (S)) = 0$ and $Q (D) = E_{S} (D)$ for Borel $D \subseteq σ (S)$ (Bounded normal operator abstract spectral theorem, Continuous functional calculus for bounded self adjoint operators, Continuous functional calculus produces a regular PVM, Borel functional calculus for a bounded normal operator, Support and uniqueness of the spectral measure, Self-adjoint, positive, unitary and normal operators).

[F4]

For an abelian concrete von Neumann algebra $A$ on a nonzero separable $H$ and a bounded self-adjoint generator $S \in A$ with $A = W^{*} (S)$ , the spectral multiplicity construction produces a nonzero finite regular Borel measure $ν$ on $K = σ (S) \subseteq R$ , a Borel multiplicity $m : K \to {1, 2, \dots} \cup {\infty}$ , and a unitary $U : H \to \int_{K}^{\oplus} C^{m (t)} d ν (t)$ with $U S U^{- 1} = M_{t}$ and $U A U^{- 1} = {M_{f} : f \in L^{\infty} (K, ν)}$ ; the construction (the cited proof's steps 1.2–1.9 and 2.1) uses only that $S$ is a prescribed bounded self-adjoint generator, its initial selection step being immaterial for a given $S$ ; for a fixed generator the measure class and multiplicity function are unique (Spectral multiplicity model for separably acting abelian von Neumann algebras, Von Neumann algebras and commutants, Direct integral of a measurable Hilbert field).

[F5]

In the model of [F4] the fibre $C^{m (t)}$ is nonzero for every $t$ and $ν$ is faithful for $E_{S}$ : $ν (N) = 0$ iff $E_{S} (N) = 0$ , because multiplication by $1_{N}$ is the zero operator exactly when the indicator vanishes almost everywhere (Spectral multiplicity model for separably acting abelian von Neumann algebras, Measurable Hilbert field from a countable fundamental family, Direct integrals of measurable Hilbert fields are Hilbert spaces).

[F6]

Finite Borel measures on the second-countable LCH space $[0, 1]$ are regular; the Radon–Nikodym theorem gives densities for mutually absolutely continuous finite Borel measures and the corresponding isometry of $L^{2}$ spaces intertwines multiplication operators (Locally finite Borel measures on second-countable LCH spaces are regular, A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density).

[F7]

Direct integrals transport along bimeasurable base isomorphisms, and multiplication operators transport accordingly (Direct integrals transport along bimeasurable base isomorphisms).

[F8]

The commutant of the diagonal multiplications on a direct integral consists of the decomposable operators; measurable sections and operator fields obey the usual calculus (Decomposable operators are the commutant of diagonal multiplication, Measurable and decomposable operator fields, Measurable essentially bounded operator fields act decomposably, Measurable sections have measurable pointwise inner products, Composition with a Borel measurable outer map preserves measurability).

[F9]

AC is the standing hypothesis (The Axiom of Choice, Separability: the existence of an at most countable dense subset, Hilbert space).

Proof

technique · direct

Given: AC, the PVM $P$ , the separable nonzero $H$ , and a finite $P$ -faithful measure $μ$ ; also the density construction of the statement.

1.1F1

Existence of a faithful measure: for a dense sequence $(ξ_{j})$ with $ξ_{j} \neq 0$ put $μ_{0} = \sum_{j} 2^{- j} ∥ ξ_{j} ∥^{- 2} E_{ξ_{j}, ξ_{j}}$ . This is a finite Borel measure, and $μ_{0} (E) = 0$ forces $E_{ξ_{j}, ξ_{j}} (E) = ⟨ P (E) ξ_{j}, ξ_{j} ⟩ = 0$ for every $j$ , so $P (E) ξ_{j} = 0$ for every $j$ ; density of $(ξ_{j})$ and boundedness of $P (E)$ give $P (E) = 0$ ; the converse is immediate. Hence $μ_{0}$ is $P$ -faithful. If $μ_{1}, μ_{2}$ are $P$ -faithful, then $μ_{1} (E) = 0 \Leftrightarrow P (E) = 0 \Leftrightarrow μ_{2} (E) = 0$ , so they are mutually absolutely continuous.

1.2F1F2F3F6

Let $c : X \to [0, 1]$ be a bimeasurable injection onto the Borel set $Y = c (X)$ by [F2], and put $S = \int_{X} c d P$ , a bounded self-adjoint operator by [F3]. Then $Q (D) : = P (c^{- 1} (D))$ for Borel $D \subseteq [0, 1]$ is a projection-valued measure on $[0, 1]$ , because $D \mapsto c^{- 1} (D)$ preserves the Boolean operations: $Q (\emptyset) = 0$ , $Q ([0, 1]) = P (X) = I$ , $Q (D_{1}) Q (D_{2}) = P (c^{- 1} (D_{1}) \cap c^{- 1} (D_{2})) = Q (D_{1} \cap D_{2})$ , and countable additivity transfers. Its coordinate integral is $S$ : by [F1] and change of variables for the PVM, $\int_{[0, 1]} z d Q (z) = \int_{[0, 1]} z d P (c^{- 1} (z)) = \int_{X} c (x) d P (x) = S$ . For $z \notin [0, 1]$ , the bounded integral of $(z - t)^{- 1}$ against $Q$ is a two-sided inverse of $z I - S$ by [F1], so $σ (S) \subseteq [0, 1]$ . Each scalar measure of $Q$ is a finite Borel measure on the compact metric space $[0, 1]$ , hence regular by [F6], so $Q$ is a regular PVM.

2.1F3step 1.2

Spectral identification: by the uniqueness clause of [F3] applied to the regular PVM $Q$ on $Λ = [0, 1]$ , one has $Q ([0, 1] ∖ σ (S)) = 0$ and $Q (D) = E_{S} (D)$ for every Borel $D \subseteq σ (S)$ . Extend $E_{S}$ by zero outside $σ (S)$ when writing it on $[0, 1]$ . Consequently, since $c^{- 1} (c (B)) = B$ , $P (B) = Q (c (B)) = E_{S} (c (B)) (B \subseteq X Borel),$ with $c (B)$ Borel by bimeasurability; in particular $E_{S}$ is carried by $Y \cap σ (S)$ because $E_{S} (c (X)) = P (X) = I$ . Set $A : = W^{*} (S)$ ; it is abelian because polynomials in the self-adjoint $S$ commute and commutation with a fixed bounded operator is WOT closed, so their WOT closure still commutes pairwise. Thus $S$ is a prescribed self-adjoint generator to which [F4] applies.

3.1F4F5step 2.1

Apply the spectral multiplicity model of [F4] to the pair $(A, S)$ : there are a finite regular Borel measure $ν$ on $K = σ (S)$ , a Borel function $m : K \to {1, 2, \dots} \cup {\infty}$ and a unitary $U : H \to \int_{K}^{\oplus} C^{m (t)} d ν (t)$ with $U S U^{- 1} = M_{t}$ and $U A U^{- 1} = {M_{f}}$ . Since $U 1_{D} (S) U^{- 1} = 1_{D} (M_{t}) = M_{1_{D}}$ , the operator $U E_{S} (D) U^{- 1}$ is the multiplication by $1_{D}$ , so $ν$ is $E_{S}$ -faithful: $ν (D) = 0$ iff $M_{1_{D}} = 0$ iff $U E_{S} (D) U^{- 1} = 0$ iff $E_{S} (D) = 0$ , the middle equivalence using that every fibre is nonzero so that a multiplication operator is zero exactly when its symbol vanishes a.e.

4.1F6step 2.1step 3.1

The pushforward $ν_{0} : = c_{*} μ$ restricted to $Y$ is likewise $E_{S}$ -faithful: for Borel $D \subseteq [0, 1]$ one has $ν_{0} (D) = μ (c^{- 1} D) = 0$ iff $P (c^{- 1} D) = E_{S} (c (c^{- 1} D)) = E_{S} (D \cap Y) = E_{S} (D)$ , using $P$ -faithfulness of $μ$ and $E_{S}$ being carried by $Y$ from [step 2.1]. Extend $ν$ by zero off $K$ , restrict both measures to $Y$ , and extend $m$ by $1$ on $Y ∖ K$ , which is null. Identify the integrals over $K$ and $Y$ by restriction and zero extension, since $ν (K ∖ Y) = 0$ . Hence $ν$ and $ν_{0}$ are mutually absolutely continuous finite Borel measures on the standard Borel space $Y$ and have Radon–Nikodym densities; the isometry $J : L^{2} (Y, ν_{0}; C^{m}) \to L^{2} (Y, ν; C^{m})$ , $J ξ = \sqrt{d ν_{0} / d ν} ξ$ , is unitary and commutes with every bounded Borel scalar multiplier by [F6]. Thus $U^{'} : = J^{- 1} \circ U$ is a unitary $H \to \int_{Y}^{\oplus} C^{m (t)} d ν_{0} (t)$ with $U^{'} A U^{' - 1} = {M_{f}}$ and $U^{'} S U^{' - 1} = M_{t}$ .

5.1F7F8step 3.1step 4.1

Transport along $c$ : the map $c : X \to Y$ is a bimeasurable bijection with $c_{*} μ = ν_{0}$ , so by [F7] pullback of sections is a unitary $T : \int_{Y}^{\oplus} C^{m (t)} d ν_{0} (t) \to \int_{X}^{\oplus} C^{m (c (x))} d μ (x)$ intertwining multiplication by $f$ with multiplication by $f \circ c$ . Define $m^{'} (x) : = m (c (x))$ , a Borel function $X \to {1, 2, \dots} \cup {\infty}$ by [F8].

6.1step 2.1step 5.1step 4.1

The composite $W : = T \circ U^{'}$ is a unitary $H \to \int_{X}^{\oplus} C^{m^{'} (x)} d μ (x)$ , and for every Borel $B \subseteq X$ , $W P (B) W^{- 1} = T U^{'} E_{S} (c (B)) U^{' - 1} T^{- 1} = T M_{1_{c (B)}} T^{- 1} = M_{1_{c (B)} \circ c} = M_{1_{B}}$ , using $P (B) = E_{S} (c (B))$ from [step 2.1] and the intertwining property of $T$ .

7.1step 1.1step 3.1step 5.1step 6.1F9∎

Steps 1.1, 5.1 and 6.1 produce the faithful finite measure $μ$ , the Borel multiplicity $m^{'}$ and the unitary $W$ with $W P (E) W^{- 1} = M_{1_{E}}$ ; any two $P$ -faithful measures are mutually absolutely continuous by [step 1.1]. The uniqueness of the multiplicity is the rigidity statement of the intertwiner lemma: two models over the same base with a unitary intertwining all multiplications have the same multiplicity almost everywhere, and such an intertwiner is decomposable with unitary fibres a.e. (Unitary intertwiners preserve fibre multiplicity over a standard Borel base).

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Ergodic systems with regular orbits concentrate on one orbit

Statement

Assume AC. Let $(U, P)$ be an ergodic system of imprimitivity for a Borel action of a group $G$ on a standard Borel space $X$ , acting on a nonzero separable Hilbert space, and suppose the orbit equivalence relation of the action is regular: there is a countable family $E_{1}, E_{2}, \dots$ of $G$ -invariant Borel subsets of $X$ such that every orbit is the intersection of the sets $E_{n}$ that contain it. Equivalently, some countable family of invariant Borel sets separates distinct orbits, which is the condition that the orbit space is countably separated. Then there is an orbit $C \subseteq X$ with $P (X ∖ C) = 0$ .

Facts & Assumptions

Given: AC, the Borel $G$ -space $X$ , the ergodic system of imprimitivity $(U, P)$ on a nonzero separable Hilbert space, and a countable family $(E_{n})$ of invariant Borel sets as in the statement.

[F1]

$(U, P)$ is a strongly continuous unitary representation together with a PVM with $U_{g} P (E) U_{g}^{- 1} = P (g E)$ , and ergodicity means that every Borel $E$ with $U_{g} P (E) U_{g}^{- 1} = P (E)$ for all $g$ satisfies $P (E) = 0$ or $P (E) = I$ ; for invariant Borel $E$ one has $P (E)$ invariant (Systems of imprimitivity for a Borel $G$ -space, Left group actions, transitive actions, and faithful actions, A measurable function between measurable spaces).

[F2]

Projections in the range of a PVM satisfy $P (A) P (B) = P (A \cap B)$ ; a projection $P (A)$ is zero exactly when the scalar measures $E_{x} (A) = ⟨ P (A) x, x ⟩$ vanish for all $x$ ; and $P$ is strongly countably additive (Scalar and complex measures from a pvm, Bounded borel pvm integral).

[F3]

$X$ is standard Borel, so Borel sets are closed under countable unions and intersections; invariance of a Borel set $E$ means $g E = E$ for all $g$ and implies invariance of its complement (Standard Borel spaces).

[F4]

AC is the standing hypothesis; it is inherited from the ambient system (The Axiom of Choice, Multiplicity model of a projection-valued measure over a standard Borel base).

Proof

technique · direct

Given: AC, the ergodic system $(U, P)$ and the countable invariant family $(E_{n})$ .

1.1F3

The two regularity formulations are equivalent. If every orbit is the intersection of the invariant Borel sets containing it, the family $(E_{n})$ separates distinct orbits: if $x, y$ lie in different orbits and $y$ belonged to every $E_{n}$ containing $x$ , then $y$ would lie in the intersection defining the orbit of $x$ . Conversely, adjoin the complements to a countable separating family and reenumerate it as $(E_{n})$ . Then for every $x$ the intersection $I_{x} = ⋂ {E_{n} : x \in E_{n}}$ is contained in the orbit of $x$ : a point $y \notin G x$ is separated from $x$ by some $E_{n}$ , and replacing $E_{n}$ by its complement if necessary (also invariant Borel by [F3]) gives $E_{n} ∋ x$ , $E_{n} ∌ y$ ; the reverse inclusion holds because each $E_{n}$ is invariant.

1.2F1

Each $P (E_{n})$ is $0$ or $I$ : since $E_{n}$ is invariant, $U_{g} P (E_{n}) U_{g}^{- 1} = P (g E_{n}) = P (E_{n})$ for every $g$ , so ergodicity applies.

2.1F2F3step 1.2

Define $F_{n} = E_{n}$ if $P (E_{n}) = I$ and $F_{n} = X ∖ E_{n}$ if $P (E_{n}) = 0$ ; each $F_{n}$ is invariant Borel and $P (F_{n}) = I$ . For $C : = ⋂_{n} F_{n}$ one has $X ∖ C = ⋃_{n} (X ∖ F_{n})$ with $P (X ∖ F_{n}) = 0$ ; strong countable additivity gives $P (⋃_{n} (X ∖ F_{n})) = \lim_{N} P (⋃_{n \leq N} (X ∖ F_{n}))$ and the scalar measures of each finite union vanish, so $P (X ∖ C) = 0$ .

3.1step 1.1step 2.1F3

$C$ is nonempty because $P (C) = I \neq 0$ on the nonzero Hilbert space, and $C$ is invariant. Choose $x \in C$ . Its orbit satisfies $G x \subseteq C$ since $C$ is invariant. Conversely, if $y \in C$ and $E_{n} ∋ x$ , then either $F_{n} = E_{n}$ , in which case $y \in C \subseteq E_{n}$ ; or $F_{n} = X ∖ E_{n}$ and $x \notin E_{n}$ , a contradiction. Hence $y$ belongs to every $E_{n}$ containing $x$ , so by regularity $y \in G x$ . Therefore $C = G x$ is a single orbit and is Borel as a countable intersection.

4.1step 2.1step 3.1F4∎

Combining [step 2.1] and [step 3.1]: the invariant Borel set $C$ is exactly one orbit and $P (X ∖ C) = 0$ , which is the concentration claim.

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Haar null classes and Borel descent on a homogeneous space

Statement

Assume AC. Let $G$ be second-countable LCH, $H$ closed, $q : G \to G / H$ , and $s$ a Borel section. Every nonzero $σ$ -finite quasi-invariant Borel measure $μ$ on $G / H$ is equivalent to a rho-derived quotient measure. The coordinates $(x, h) \mapsto s (x) h$ carry the product quotient/Haar measure class to the Haar measure class on $G$ . In particular $μ (E) = 0$ iff $q^{- 1} (E)$ is Haar null. If a Borel map $F : G / H \times H \to U (K)$ for separable $K$ satisfies $F (x, h k) = F (x, h)$ for every $k$ and almost every $(x, h)$ , then $F (x, h) = B (x)$ almost everywhere for a Borel $B : G / H \to U (K)$ .

Facts & Assumptions

Given: AC, the second-countable LCH group $G$ , the closed subgroup $H$ , the quotient map $q$ , and a Borel section $s$ with its cocycle as in Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups.

[F1]

The map $Θ : G / H \times H \to G$ , $Θ (x, h) = s (x) h$ , is a Borel isomorphism with Borel inverse $g \mapsto (q (g), s (q (g))^{- 1} g)$ , and $q^{- 1} (E) = Θ (E \times H)$ for every $E \subseteq G / H$ (Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

[F2]

Fix left Haar measures on $G$ and $H$ and a rho-function $ρ > 0$ , continuous, with $ρ (x h) = Δ_{H} (h) Δ_{G} (h)^{- 1} ρ (x)$ . The Weil formula $\int_{G} f (t) ρ (t) d t = \int_{G / H} \int_{H} f (x h) d h d μ_{ρ} (x H)$ holds for $f \in C_{c} (G)$ ; $μ_{ρ}$ is a full-support nonzero Radon measure that is strongly quasi-invariant, and $ρ d t$ is a Radon measure equivalent to Haar (Weil formula with a rho-function, Existence of rho-functions and quotient measure classes, Rho-function for a closed subgroup, Quasi-invariant Radon measure on G/H).

[F3]

Every Borel measure finite on compact sets on the second-countable LCH space $G$ is regular, so two such measures agreeing on $C_{c} (G)$ agree on Borel sets (Assuming Dependent Choice, uniqueness of the RMK representing measure among Radon measures, Radon measure on an LCH space, Locally finite Borel measures on second-countable LCH spaces are regular).

[F4]

Completed-product Tonelli/Fubini applies to $σ$ -finite measures and nonnegative measurable functions; the left Haar measure $d h$ is invariant under left translations on $H$ , so the homeomorphism $(h, k) \mapsto (h, h k)$ of $H \times H$ preserves the completed product $d h \otimes d h$ (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, Monotone convergence for the integral, Right translation scales left Haar measure).

[F5]

A separable $K$ has a finite or countable orthonormal basis, and matrix coefficients of operators in $U (K)$ against it are bounded Borel functions on $U (K)$ ; integration of a bounded Borel $U (K)$ -valued function against a probability density produces the matrix of a bounded operator, and the unitary conditions are countably many Borel equations (A Hilbert space with a dense sequence has a finite or countable orthonormal basis, Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F6]

AC implies DC and Countable Choice, which are the choice principles used by the Tonelli, monotone-convergence and RMK interfaces (The Axiom of Choice, AC implies DC implies countable choice).

Proof

technique · direct

Given: AC, the data of the statement, and a rho-function $ρ$ with its measure $μ_{ρ}$ .

1.1F2F3

Extension of the Weil formula to Borel sets: for every Borel $E \subseteq G$ , $\int_{G / H} \int_{H} 1_{E} (x h) d h d μ_{ρ} (x) = \int_{E} ρ (t) d t .$ Both sides are $σ$ -finite Borel measures on the $σ$ -compact space $G$ that are finite on compacta (the right side because $ρ$ is continuous; the left side by sandwiching indicators of a compact set between $C_{c}$ functions). They agree on $C_{c} (G)$ by [F2], so by [F3] they agree on every Borel set.

1.2F4

Descent, first reduction: let $F$ be as in the statement. The set $N = {(x, h, k) : F (x, h k) \neq F (x, h)}$ is a Borel subset of the triple product, and its measure is zero: by Tonelli its measure is the integral over $k$ of the measures of the sections $N_{k} = {(x, h) : F (x, h k) \neq F (x, h)}$ , each of which is null by hypothesis. Hence Fubini gives that for a.e. $x$ the section $N_{x}$ is a null subset of $H \times H$ .

2.1step 1.1F1F2

The coordinate map pushes the product measure to $ρ d t$ : by [step 1.1] and left invariance of $d h$ (which lets $s (x)$ be replaced by any representative of the coset in $\int_{H} 1_{E} (\cdot h) d h$ ), for every Borel $E \subseteq G$ one has $(μ_{ρ} \otimes d h) (Θ^{- 1} (E)) = \int_{G / H} \int_{H} 1_{E} (s (x) h) d h d μ_{ρ} (x) = \int_{E} ρ (t) d t$ . Since $0 < ρ < \infty$ pointwise, the classes of $ρ d t$ and $d t$ coincide; hence the product class maps to the Haar class and, by [F1], $q^{- 1} (E)$ is Haar null iff $(μ_{ρ} \otimes d h) (E \times H) = 0$ iff $μ_{ρ} (E) = 0$ .

2.2F4step 1.2

For such an $x$ , the homeomorphism $(h, k) \mapsto (h, h k)$ preserves the completed product $d h \otimes d h$ by [F4], so its image of $N_{x}$ , namely ${(h_{1}, h_{2}) : F (x, h_{2}) \neq F (x, h_{1})}$ , is null. Hence $F_{x}$ is $d h$ -a.e. constant for a.e. $x$ .

3.1F2F4step 2.1

Equivalence of two rho measures and of any quasi-invariant measure with $μ_{ρ}$ : if $μ$ is a nonzero $σ$ -finite quasi-invariant Borel measure on $G / H$ , replace it by an equivalent probability (still written $μ$ ), choose a probability $ν = w (t) d t$ with $w > 0$ Haar-a.e., and set $\overline{μ} (E) = \int_{G} μ (g^{- 1} E) d ν (g)$ for Borel $E$ . The integrand is Borel in $g$ for fixed $E$ , and $\overline{μ}$ is a probability on $G / H$ . Each translate $g_{*} μ (E) = μ (g^{- 1} E)$ is equivalent to $μ$ , so $\overline{μ} \sim μ$ ; and by Tonelli $\overline{μ} (E) = \int_{G / H} ν {g : g x \in E} d μ (x)$ . For fixed $x$ with representative $t = s (x)$ , the substitution $g \mapsto g t$ gives $ν {g : g x \in E} = Δ_{G} (t)^{- 1} \int_{q^{- 1} (E)} w (u t^{- 1}) d u$ by the right-translation scaling of the Haar integral; since $w > 0$ Haar-a.e. and $q^{- 1} (E)$ is right- $H$ -invariant, this is positive exactly when $μ_{ρ} (E) > 0$ by [step 2.1]. Hence $\overline{μ} (E) = 0$ iff $μ_{ρ} (E) = 0$ , so $μ \sim μ_{ρ}$ . This proves the first assertion and, with [step 2.1], the null-class assertion $μ (E) = 0 \Leftrightarrow q^{- 1} (E)$ Haar null.

3.2F5step 2.2

Borel selection of the constant: fix a strictly positive integrable Borel probability density $q$ on $H$ : take a countable compact cover $(K_{n})$ , each of finite Haar measure, and normalize $\sum_{n} 2^{- n} (1 + ∣ K_{n} ∣)^{- 1} 1_{K_{n}}$ , which is positive everywhere and has finite nonzero integral and a finite or countable orthonormal basis $(e_{i})$ of $K$ by [F5]; set $b_{i j} (x) = \int_{H} ⟨ F (x, h) e_{i}, e_{j} ⟩ q (h) d h$ . Each $b_{i j}$ is Borel in $x$ by Tonelli, and for a.e. $x$ the matrix $(b_{i j} (x))$ is the matrix of the a.e. constant unitary value of $F_{x}$ , hence unitary. The set of $x$ where the countably many Borel unitary equations $\sum_{j} b_{i j} \overline{b_{i^{'} j}} = δ_{i i^{'}}$ and column completeness fail is Borel and null; put $B (x) = I$ there and $B (x)$ equal to the operator with matrix $(b_{i j} (x))$ otherwise. Then $B : G / H \to U (K)$ is Borel and $F (x, h) = B (x)$ for a.e. $(x, h)$ .

4.1step 2.1step 3.1step 3.2F6∎

Collecting the steps: every nonzero $σ$ -finite quasi-invariant $μ$ is equivalent to $μ_{ρ}$ [step 3.1]; the coordinate map carries the product class to the Haar class, so a Borel $E \subseteq G / H$ is $μ$ -null exactly when its full preimage $q^{- 1} (E)$ is Haar null [step 2.1, step 3.1]; and every $H$ -invariant Borel $U (K)$ -valued function descends to a Borel $B$ almost everywhere [step 3.2]. These are the three assertions of the statement.

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Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity

Statement

Assume AC, and let $N$ be a second-countable locally compact Hausdorff abelian group, $π : N \to U (H)$ a strongly continuous unitary representation on a separable Hilbert space $H$ , and $K$ a second-countable locally compact group acting continuously on $N$ by automorphisms $α_{k}$ , with dual action $k \cdot χ = χ \circ α_{k}^{- 1}$ on $\hat{N}$ . If $τ : K \to U (H)$ is a strongly continuous unitary representation with $τ (k) π (n) τ (k)^{- 1} = π (α_{k} (n))$ , then there is a unique regular projection-valued measure $P$ on $\hat{N}$ with $π (n) = \int_{\hat{N}} χ (n) d P (χ) (n \in N), τ (k) P (E) τ (k)^{- 1} = P (k \cdot E)$ for every Borel $E \subseteq \hat{N}$ . If in addition the representation $n \mapsto π (n)$ , $k \mapsto τ (k)$ of $N ⋊ K$ is irreducible, then the measure class of $P$ is ergodic for the action of $K$ on $\hat{N}$ : every Borel $E$ with $P (E)$ invariant under $τ$ satisfies $P (E) = 0$ or $P (E) = I$ .

Facts & Assumptions

Given: AC, the groups $N, K$ , the strongly continuous representations $π, τ$ with the covariance relation, and a separable $H$ .

[F1]

$L^{1} (N)$ is a commutative complex Banach $*$ -algebra with convolution and involution $f^{*} (n) = Δ_{N} (n)^{- 1} \overline{f (n^{- 1})}$ ; $C_{c} (N)$ is dense; there is a contractively bounded approximate identity; the Haar integral satisfies the inversion formula $\int_{G} w (n) d n = \int_{G} Δ_{G} (m)^{- 1} w (m^{- 1}) d m$ ; nonnegative compactly supported functions exist near every point and Haar measure is positive on nonempty open sets (L1 of a locally compact group is a Banach star-algebra, Involution on L1 of a locally compact group, Haar change of variables under inversion, L1 group algebras have a contractively bounded approximate identity, Convolution on L1 of a locally compact group, Complex Haar L^p spaces and compactly supported functions, LCH Urysohn cutoff, Haar measure is positive on nonempty open sets and finite on compact sets).

[F2]

Bochner calculus in $H$ : strong measurability plus finiteness of $\int ∥ F ∥$ gives Bochner integrability; $∥ \int F ∥ \leq \int ∥ F ∥$ ; bounded linear maps commute with the Bochner integral; norm dominated convergence holds; scalar Fubini applies to iterated integrals of integrable kernels (Bochner integrability criterion, Bochner integral norm inequality, Bounded linear maps commute with Bochner integration, Bochner dominated convergence theorem, Fubini's theorem for L^1 functions on a sigma-finite product, Bochner-integrable function).

[F3]

Every nonzero complex-linear multiplicative functional on $L^{1} (N)$ is $λ_{χ} (f) = \int f χ d n$ for a unique $χ \in \hat{N}$ , and the Fourier transforms $\hat{f}$ form a self-adjoint separating algebra with uniform closure $C_{0} (\hat{N})$ ; $\hat{N}$ is locally compact abelian (Characters of the L1 algebra of an abelian group, LCA Fourier transforms form a dense algebra in C0 of the dual, The dual of a locally compact abelian group is locally compact abelian, The Pontryagin dual with the compact-open topology).

[F4]

For a commutative C*-algebra $A$ , the Gelfand transform is an isometric $*$ -isomorphism onto $C_{0} (Δ (A))$ , so $∥ a ∥ = \sup_{ψ \in Δ (A)} ∣ ψ (a) ∣$ (Nonunital commutative Gelfand Naimark, Gelfand transform).

[F5]

A nondegenerate star representation $T : C_{0} (X) \to B (H)$ on a separable Hilbert space is $T (g) = \int_{X} g d P$ for a unique regular PVM $P$ with $P (X) = I$ (Nondegenerate representations of C0 have regular PVMs, Projection valued measure).

[F6]

PVM integral calculus: $Φ_{P} (g) = \int g d P$ is a unital $*$ -homomorphism of bounded Borel functions, $∥ Φ_{P} (g) ∥ \leq ∥ g ∥_{\infty}$ , scalar measures $P_{ξ, η}$ are finite complex measures with $∣ P_{ξ, η} ∣ (X) \leq ∥ ξ ∥ ∥ η ∥$ , and bounded pointwise convergence gives strong convergence (Bounded borel pvm integral, Pvm integral is a star homomorphism, Scalar and complex measures from a pvm, Dominated convergence).

[F7]

$N$ is Polish and a countable union of compacta, so Haar measure is $σ$ -finite. Its scalar $L^{2}$ space is separable by the direct-integral Hilbert-space theorem; step 1.1 derives $L^{1}$ separability and hence second countability of the dual (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Second countability: an at most countable basis for the topology, Left Haar integral and left Haar measure, Direct integrals of measurable Hilbert fields are Hilbert spaces).

[F8]

AC implies DC and Countable Choice for the Bochner, Fubini and Gelfand interfaces (The Axiom of Choice, AC implies DC implies countable choice, Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Hilbert space).

Proof

technique · direct

Given: AC, $N, K, π, τ$ and the covariance relation.

1.1F3F7F8

If $H = 0$ , the zero PVM uniquely satisfies the statement, and the irreducibility premise does not hold. Assume $H \neq 0$ . Let $C_{j}$ be increasing compact sets covering $N$ . The closed subspace of $L^{2} (N)$ supported in $C_{j}$ is separable by [F7], and its inclusion into $L^{1} (N)$ is continuous, with norm at most $μ (C_{j})^{1 / 2}$ . Choosing a countable dense family in each such subspace gives a countable $L^{1}$ -dense family: for any $f \in L^{1}$ , the truncations $f 1_{C_{j}} 1_{∣ f ∣ \leq r}$ lie in these $L^{2}$ subspaces and converge to $f$ in $L^{1}$ as $j, r \to \infty$ . Thus $L^{1} (N)$ is separable. On its dual unit ball, evaluation on a countable norm-dense family induces the pointwise-evaluation topology, because the norm bounds uniformly control the error of replacing any argument by a dense one. This embeds that ball into a countable product of complex lines. By [F3] the dual $\hat{N}$ is homeomorphic to its character subspace and therefore second countable; since it is LCH, it is standard Borel by [F7].

2.1F1F2F7step 1.1

For $f \in L^{1} (N)$ define $π (f) ξ = \int_{N} f (n) π (n) ξ d n$ . The integrand is strongly measurable (a.e. limit of $C_{c}$ -approximants times the continuous map $n \mapsto π (n) ξ$ ) and $\int_{N} ∥ f (n) π (n) ξ ∥ d n = ∥ f ∥_{1} ∥ ξ ∥ < \infty$ , so $π (f)$ is a bounded operator with $∥ π (f) ∥ \leq ∥ f ∥_{1}$ ; $f \mapsto π (f)$ is linear and multiplicative: for $f, g \in C_{c} (N)$ Fubini, applicable since the Haar measure is $σ$ -finite by [F7], gives $π (f) π (g) = \int_{N} \int_{N} f (m) g (n) π (m n) d m d n = π (f * g)$ , and both sides extend by density; and $π (f)^{*} = π (f^{*})$ by the adjoint computation and the inversion formula of [F1]. Nondegeneracy: for the approximate identity $e_{U}$ one has $π (e_{U}) ξ \to ξ$ because $∥ π (e_{U}) ξ - ξ ∥ \leq \int e_{U} (n) ∥ π (n) ξ - ξ ∥ d n$ and strong continuity makes the integrand small on $supp e_{U}$ eventually.

3.1F3F4step 2.1

Let $A$ be the norm closure of $π (L^{1} (N))$ in $B (H)$ ; it is a commutative C*-algebra (the image of the commutative $L^{1} (N)$ is a commutative $*$ -algebra) and it is nonzero when $H \neq 0$ by [step 2.1]. For $y \in Δ (A)$ the composite $y \circ π$ is a nonzero complex-linear multiplicative functional on $L^{1} (N)$ : if it vanished on the dense subalgebra $π (L^{1} (N))$ then $y = 0$ by continuity. Hence by [F3] there is $χ \in \hat{N}$ with $y (π (f)) = \hat{f} (χ)$ ; therefore, using the isometry of [F4], $∥ π (f) ∥ = \sup_{y \in Δ (A)} ∣ y (π (f)) ∣ \leq \sup_{χ \in \hat{N}} ∣ \hat{f} (χ) ∣ = ∥ \hat{f} ∥_{\infty}$ .

4.1F3step 2.1step 3.1

The assignment $\hat{f} \mapsto π (f)$ is well defined and linear because $\hat{f} = 0$ forces $π (f) = 0$ by [step 3.1], and it is bounded for the uniform norm; since the Fourier transforms are uniformly dense in $C_{0} (\hat{N})$ by [F3], it extends uniquely to a bounded linear map $T : C_{0} (\hat{N}) \to B (H)$ with $T (\hat{f}) = π (f)$ . Multiplicativity and $*$ -preservation extend from the dense subalgebra of Fourier transforms, using continuity of the products, so $T$ is a nondegenerate star representation: $T (C_{0} (\hat{N})) H$ is dense because $T ({\hat{e}}_{U}) ξ = π (e_{U}) ξ \to ξ$ for the approximate identity.

5.1F5step 4.1

By [F5] there is a unique regular PVM $P$ on $\hat{N}$ with $T (g) = \int_{\hat{N}} g d P$ for all $g \in C_{0} (\hat{N})$ ; in particular $π (f) = \int_{\hat{N}} \hat{f} d P$ for every $f \in L^{1} (N)$ , and $P (\hat{N}) = I$ .

6.1F6F7step 5.1

Put $R (n) = \int_{\hat{N}} χ (n) d P (χ)$ , using the PVM just constructed. The PVM calculus gives $R (n) R (m) = R (n m)$ and $R (n)^{*} R (n) = R (n) R (n)^{*} = I$ . For every sequence $n_{j} \to n_{0}$ , $∥ (R (n_{j}) - R (n_{0})) ξ ∥^{2} = \int ∣ χ (n_{j}) - χ (n_{0}) ∣^{2} d P_{ξ} \to 0$ by dominated convergence; since $N$ is metrizable, this proves strong continuity. The zero Hilbert space has the zero PVM throughout, so the same conclusions hold there.

6.2F1F2F6F7step 5.1

For $f \in L^{1} (N)$ one has $\int_{N} f (n) R (n) d n = π (f)$ : the evaluation $(n, χ) \mapsto χ (n)$ is jointly continuous: restrict $n$ to a compact neighbourhood of $n_{0}$ and use uniform convergence of characters there together with continuity of the limiting character. Thus the kernel below is jointly Borel. Pairing with $η$ and commuting the bounded functional through the Bochner integral, $⟨ \int_{N} f (n) R (n) d n ξ, η ⟩ = \int_{N} f (n) \int_{\hat{N}} χ (n) d P_{ξ, η} (χ) d n$ , and Fubini, applied to the product of the $σ$ -finite Haar measure and the finite measure $P_{ξ, η}$ by [F7], identifies this with $\int_{\hat{N}} \hat{f} d P_{ξ, η} = ⟨ π (f) ξ, η ⟩$ by [step 5.1]. Hence the continuous function $h (n) = ⟨ π (n) ξ, η ⟩ - ⟨ R (n) ξ, η ⟩$ satisfies $\int_{N} f h d n = 0$ for every $f \in L^{1} (N)$ ; if $h (n_{0}) \neq 0$ , rotate $h$ by a scalar of modulus one so its value at $n_{0}$ has positive real part; a nonnegative compactly supported cutoff supported where that real part remains positive has a nonzero integral against $h$ , a contradiction, so $h = 0$ . As $ξ, η$ were arbitrary, $π (n) = R (n)$ for every $n$ , i.e. $π (n) = \int_{\hat{N}} χ (n) d P (χ)$ .

7.1F3F5F6step 6.2

Uniqueness of $P$ : if $P^{'}$ is another regular PVM on $\hat{N}$ with $π (n) = \int χ (n) d P^{'} (χ)$ for all $n$ , then for every $f \in L^{1} (N)$ , $\int \hat{f} d P^{'} = \int f (n) π (n) d n$ as above, so $\int g d P^{'} = \int g d P$ for all $g$ in the uniformly dense algebra of Fourier transforms; both sides are bounded linear in $g \in C_{0} (\hat{N})$ , so the equality holds on all of $C_{0} (\hat{N})$ , and [F5] applied to the common representation gives $P^{'} = P$ .

8.1F3step 6.2step 7.1

Covariance: fix $k \in K$ . The map $χ \mapsto k \cdot χ$ is a homeomorphism of $\hat{N}$ , so $Q (E) : = P (k \cdot E)$ is a regular PVM, and $P_{k} (E) : = τ (k)^{- 1} Q (E) τ (k)$ is again a regular PVM. Its integrated representation is $\int χ (n) d P_{k} (χ) = τ (k)^{- 1} \int χ (n) d P (k \cdot χ) τ (k) = τ (k)^{- 1} π (α_{k} (n)) τ (k) = π (n)$ for every $n$ , where the change of variables in the dual and the covariance relation were used. By [step 7.1] $P_{k} = P$ , that is $τ (k) P (E) τ (k)^{- 1} = P (k \cdot E)$ .

9.1F6step 6.2step 8.1

Ergodicity: suppose $E$ is Borel and $P (E)$ is invariant under $τ$ , $τ (k) P (E) τ (k)^{- 1} = P (E)$ for all $k$ . For every $n$ , $π (n) P (E) = P (E) π (n)$ , since $P (E)$ is a spectral projection of $P$ and $π (n) = \int χ (n) d P$ . Hence the range of $P (E)$ is a closed subspace invariant under both $π (N)$ and $τ (K)$ ; if the semidirect-product representation is irreducible, $P (E) = 0$ or $P (E) = I$ . This is precisely the ergodicity of the measure class of $P$ for the dual action.

10.1step 5.1step 7.1step 8.1step 9.1F8∎

Steps 5.1, 7.1 and 8.1 give existence, uniqueness and covariance of the regular PVM $P$ , and [step 9.1] gives ergodicity under irreducibility; the zero-dimensional case $H = {0}$ is the zero PVM and is immediate.

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Transitive systems of imprimitivity and their normalized measure class

Definition

A system of imprimitivity $(U, P)$ on a standard Borel $G$ -space $X$ (Systems of imprimitivity for a Borel $G$ -space, Standard Borel spaces) is transitive when $X$ is $G$ -equivariantly isomorphic to a homogeneous space $G / H$ with $H \leq G$ closed, the isomorphism carrying the Borel structure of $G / H$ (Left group actions, transitive actions, and faithful actions, Left and right cosets $g H$ and $H g$ of a subgroup, Topological group: multiplication and inversion are continuous); hence then $G$ is second-countable locally compact, the action on $G / H$ is the left-coset action, and the stabilizer of the identity coset is $H$ . For a transitive system one fixes the base identification $X = G / H$ .

Assume AC for the following normalized-measure existence and uniqueness assertions: a normalized representative is a strongly quasi-invariant Radon measure $μ_{ρ}$ on $G / H$ built from a rho-function $ρ$ (Rho-function for a closed subgroup, Quasi-invariant Radon measure on G/H, Existence of rho-functions and quotient measure classes). The normalized homogeneous measure class is the unique class of nonzero quasi-invariant Radon measures on $G / H$ ; the system is called transitive on $G / H$ .

Well-definedness. The equivariant-isomorphism clause is a condition on the given system and selects the conjugacy class of $H$ : if $x_{0} \in X$ is the image of the identity coset under a $G$ -equivariant Borel isomorphism, then ${Stab}_{G} (x_{0}) = H$ by the computation $g H = H ⟺ g \in H$ (Left and right cosets $g H$ and $H g$ of a subgroup); conversely a homogeneous space $G / H$ for second-countable locally compact $G$ and closed $H$ is a standard Borel $G$ -space with Borel action (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Compact lifts and averaging onto C_c(G/H)). The normalized class exists and is unique: the rho-function theorem supplies a full-support strongly quasi-invariant Radon representative $μ_{ρ}$ (Existence of rho-functions and quotient measure classes), two rho-functions give representatives in the same class (their densities $ρ_{1} / ρ_{2}$ are positive continuous), and every nonzero $σ$ -finite quasi-invariant Borel measure is equivalent to $μ_{ρ}$ (Haar null classes and Borel descent on a homogeneous space); in particular the class does not depend on the chosen rho-function, on the normalization of Haar measure, or on the choice of the base-point identification. Consumers that use only the definitional term transitive do not consume the normalized-measure existence assertion.

The definition names no choice; AC is used exactly by the quoted rho-function and Haar-lift suppliers for existence and uniqueness of the normalized class.

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A transitive Borel $G$ -space with a quasi-invariant measure class is ergodic

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ a closed subgroup, and $μ$ a nonzero quasi-invariant Radon measure on $G / H$ . If $E \subseteq G / H$ is Borel with $μ (E △ g E) = 0$ for every $g \in G$ , then $μ (E) = 0$ or $μ ((G / H) ∖ E) = 0$ . Equivalently, any transitive system of imprimitivity on $G / H$ whose measure class is the quasi-invariant class is ergodic.

Facts & Assumptions

Given: AC, the group $G$ , closed subgroup $H$ , the quotient $q : G \to G / H$ , a nonzero quasi-invariant Radon measure $μ$ on $G / H$ , and a Borel $E \subseteq G / H$ with $μ (E △ g E) = 0$ for all $g$ .

[F1]

$q$ is continuous and open, $G / H$ is a standard Borel $G$ -space with Borel action, $q^{- 1} (g E) = g q^{- 1} (E)$ for the left action, and a Borel set $F \subseteq G / H$ is $μ$ -null if and only if $q^{- 1} (F)$ is Haar null (Compact lifts and averaging onto C_c(G/H), Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Haar null classes and Borel descent on a homogeneous space, Left group actions, transitive actions, and faithful actions).

[F2]

There exists a full-support strongly quasi-invariant rho-measure $μ_{ρ}$ whose class is the quasi-invariant class, and every nonzero quasi-invariant $σ$ -finite Borel measure is equivalent to $μ_{ρ}$ ; Radon measures on the $σ$ -compact space $G / H$ are $σ$ -finite (Existence of rho-functions and quotient measure classes, Quasi-invariant Radon measure on G/H, Haar null classes and Borel descent on a homogeneous space, Radon measure on an LCH space, Countably compact, Lindel"of, sequentially compact, limit point compact and $σ$ -compact spaces, and relatively compact subsets, Second countability: an at most countable basis for the topology, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular).

[F3]

Tonelli applies to nonnegative product-measurable functions on sigma-finite measure spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product). Right translation by $t$ scales left Haar measure by a positive scalar, so it preserves Haar-null Borel sets; this holds at the Borel-measure level, not only for $C_{c}$ integrals (Modular function of a locally compact group, Borel-level form). Left Haar measure on $G$ is sigma-finite because $G$ is sigma-compact by [F1].

[F4]

For a transitive system of imprimitivity on $G / H$ , invariance of a spectral projection $P (B)$ under the representation means $P (g B) = P (B)$ for all $g$ , and $P$ is strongly countably additive, so $P (B △ g B) = 0$ whenever $P (g B) = P (B)$ (Systems of imprimitivity for a Borel $G$ -space, Scalar and complex measures from a pvm).

[F5]

AC is the standing hypothesis (The Axiom of Choice, Topological group: multiplication and inversion are continuous).

Proof

technique · direct

Given: AC, the data and the invariant Borel set $E$ of the statement.

1.1F1F2

Lift the indicator: $f (t) : = 1_{E} (q (t))$ . For every $g$ , applying the hypothesis to $g^{- 1}$ gives $μ (E △ g^{- 1} E) = 0$ , so by [F1] the set $q^{- 1} (E △ g^{- 1} E)$ is Haar null. Since $q (g t) = g q (t)$ , one has $f (g t) = 1_{E} (g q (t)) = 1_{g^{- 1} E} (q (t))$ , which equals $f (t) = 1_{E} (q (t))$ at every $t$ outside $q^{- 1} (E △ g^{- 1} E)$ ; hence $f \circ L_{g} = f$ Haar-a.e. for every $g \in G$ .

2.1F1F3step 1.1

The function $(g, t) \mapsto ∣ f (g t) - f (t) ∣$ is Borel and hence product-measurable, since multiplication is continuous and the Borel sigma-algebra of a product of second-countable spaces is the product Borel sigma-algebra. By step 1.1 each integral in $t$ is zero. Tonelli therefore gives $\int_{G} \int_{G} ∣ f (g t) - f (t) ∣ d g d t = 0$ , so for Haar-almost every $t$ , $f (g t) = f (t)$ for Haar-almost every $g$ .

3.1F3step 2.1

Choose $t_{0}$ with the preceding property; the conull set is nonempty since Haar measure is nonzero. The exceptional set of $g$ is Haar null, and its right translate by $t_{0}$ remains null by [F3]. Substituting $u = g t_{0}$ therefore gives $f (u) = f (t_{0})$ for Haar-almost every $u$ . Since $f (t_{0}) \in {0, 1}$ , $f = 1_{q^{- 1} (E)}$ is Haar-a.e. zero or Haar-a.e. one.

4.1F1step 3.1

By the null-class equivalence of [F1], $f = 0$ Haar-a.e. gives $μ (E) = 0$ and $f = 1$ Haar-a.e. gives $μ ((G / H) ∖ E) = 0$ . This proves the first assertion.

5.1F4step 4.1F5∎

Equivalence with ergodicity of transitive systems: let $(U, P)$ be a transitive system on $G / H$ whose null class (the class of $P$ -null Borel sets) is the quasi-invariant class, and let $P (B)$ be invariant under $U$ . Then $P (g B) = P (B)$ for all $g$ , so $P (B △ g B) = 0$ by [F4] and hence $B △ g B$ is null for every measure in the quasi-invariant class; applying [step 4.1] to a representative $μ$ gives $μ (B) = 0$ or $μ (B^{c}) = 0$ , and translating back gives $P (B) = 0$ or $P (B) = I$ . Thus the system is ergodic.

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Haar regularization of transitive unitary cocycles

Statement

Assume AC. Let $G, H, μ, s$ be as in the Haar-lift lemma, $K$ separable, and $c : G \times G / H \to U (K)$ Borel with $c (g_{1} g_{2}, x) = c (g_{1}, g_{2} x) c (g_{2}, x)$ for each pair and almost every $x$ . Suppose $g \mapsto c (g, \cdot)$ is continuous in local convergence in measure in the strong topology of $U (K)$ . Then there exist a strongly continuous unitary representation $σ : H \to U (K)$ and a Borel $B : G / H \to U (K)$ with $c (g, x) = B (g x) σ (s (g x)^{- 1} g s (x)) B (x)^{- 1}$ for every $g$ and almost every $x$ . This formula gives a strict Borel cocycle on all of $G \times G / H$ . The representation $σ$ is unique up to unitary equivalence under Borel changes of fibre gauge.

Facts & Assumptions

Given: AC, a second-countable LCH group $G$ , a closed subgroup $H$ , the quotient $G / H$ with a Borel section $s$ , a nonzero quasi-invariant measure class (a representative $μ$ ), a separable Hilbert space $K$ , and a Borel cocycle $c$ .

[F1]

The Haar-lift lemma supplies: $q^{- 1} (E)$ is Haar null iff $μ (E) = 0$ ; the coordinate map $Θ (x, h) = s (x) h$ is a Borel isomorphism; and every Borel $F : G / H \times H \to U (K)$ satisfying $F (x, h k) = F (x, h)$ for all $k$ and a.e. $(x, h)$ equals $B (x)$ a.e. for a Borel $B : G / H \to U (K)$ (Haar null classes and Borel descent on a homogeneous space, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

[F2]

Steinhaus–Pettis: for a separable $K$ , $U (K)$ in the strong topology is a second-countable topological group and every Borel homomorphism $H \to U (K)$ is strongly continuous (Steinhaus and Pettis: Borel homomorphisms of second-countable locally compact groups are continuous, Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F3]

Completed-product Tonelli/Fubini for $σ$ -finite measures; left translations preserve the Haar measure and right translations scale it by the modular function; Haar null sets of the completed product are preserved by the coordinate changes used below (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, Strong continuity of left and modular right translations on L1 and L2, Monotone convergence for the integral).

[F4]

For a Borel $U (K)$ -valued function $u$ on $G$ and $ξ \in K$ , the translates $g \mapsto u (g t) ξ$ are continuous in local measure: on a fixed finite-Haar-measure set $C$ one has $\int_{C} ∥ u (g t) ξ - u (g_{0} t) ξ ∥^{2} d t \to 0$ . This follows by approximating $t \mapsto u (t) e_{j}$ on relatively compact sets by continuous compactly supported $K$ -valued functions (using a countable orthonormal basis and the density of $C_{c}$ in $L^{2}$ ) and then applying the $L^{2}$ translation continuity, uniformly over $g$ in a compact neighbourhood, where the modular factor $Δ_{G} (t)$ is bounded (Strong continuity of left and modular right translations on L1 and L2, Completeness of the complex Haar L1 and L2 spaces and density of Cc, A Hilbert space with a dense sequence has a finite or countable orthonormal basis, LCH Urysohn cutoff).

[F5]

Finite measures absolutely continuous with respect to a $σ$ -finite measure have Radon–Nikodym densities (A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density). AC implies DC and Countable Choice for the Tonelli, density and selection interfaces (The Axiom of Choice, AC implies DC implies countable choice, Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel).

Proof

technique · direct

Given: AC, the cocycle $c$ and the continuity hypothesis.

1.1F1F3

Lift to $G$ : put $C (u, v) = c (u v^{- 1}, q (v))$ , a Borel $U (K)$ -valued function on $G \times G$ . For the cocycle law applied at $g_{1} = u v^{- 1}$ , $g_{2} = v w^{- 1}$ in the second variable $q (w)$ , the equivariance $q (g x) = g q (x)$ of the quotient map gives $(v w^{- 1}) q (w) = q (v w^{- 1} w) = q (v)$ , so $C (u, v) C (v, w) = c (u v^{- 1}, (v w^{- 1}) q (w)) c (v w^{- 1}, q (w)) = c (u w^{- 1}, q (w)) = C (u, w)$ wherever the a.e. cocycle identity of $c$ holds at that triple. The set of triples $(u, v, w)$ for which it may fail is the preimage of the cocycle law's null set under the homeomorphism $(u, v, w) \mapsto (u v^{- 1}, v w^{- 1}, w)$ of $G^{3}$ , whose Jacobian is a positive modular factor; by [F3] that preimage is null. Hence $C (u, v) C (v, w) = C (u, w)$ for Haar-a.e. $(u, v, w)$ .

2.1F3step 1.1

Choose $w_{0}$ by Fubini so that $C (u, v) C (v, w_{0}) = C (u, w_{0})$ for Haar-a.e. $(u, v)$ , and set $b (u) = C (u, w_{0}) = c (u w_{0}^{- 1}, q (w_{0}))$ , a Borel $U (K)$ -valued function. Then $C (u, v) = C (u, w_{0}) C (v, w_{0})^{- 1} = b (u) b (v)^{- 1}$ for Haar-a.e. $(u, v)$ , i.e. $c (g, q (t)) = b (g t) b (t)^{- 1}$ for Haar-a.e. $(g, t)$ .

3.1F4F5step 2.1

Upgrade to every fixed $g$ : let $G_{0}$ be the conull set of $g$ for which the identity of [step 2.1] holds for a.e. $t$ ; it is dense because Haar measure is positive on nonempty open sets, so every $g_{0}$ is a limit of a net $(g_{i})$ in $G_{0}$ . Along that net the left-hand classes $t \mapsto c (g_{i}, q (t))$ converge in local measure to $t \mapsto c (g_{0}, q (t))$ by the continuity hypothesis: for a finite-Haar-measure set $C \subseteq G$ , the finite measure $q_{*} (1_{C} d t)$ is absolutely continuous with respect to $μ$ by [F1]; truncating its Radon–Nikodym density and exhausting the $σ$ -finite base shows that local convergence in $μ$ -measure implies convergence for this finite measure. Thus pullback is continuous in local Haar measure. The right-hand classes $t \mapsto b (g_{i} t) b (t)^{- 1}$ converge in local measure to $t \mapsto b (g_{0} t) b (t)^{- 1}$ by [F4]; multiplication by the fixed field $b (t)^{- 1}$ preserves this convergence, as follows by approximating $b (t)^{- 1} ξ$ on each finite-measure set by finite-valued vectors and using the uniform norm bound on unitaries. Since the two sides agree at each $g_{i}$ , uniqueness of local-measure limits gives $c (g_{0}, q (t)) = b (g_{0} t) b (t)^{- 1}$ for a.e. $t$ . As $g_{0}$ was arbitrary, the identity holds for every fixed $g$ and Haar-a.e. $t$ .

4.1F3step 3.1

Stabilizer constants: for $h \in H$ put $A_{h} (t) = b (t)^{- 1} b (t h)$ . For every $g$ and Haar-a.e. $t$ , [step 3.1] applied to $g t$ and to $t$ , together with $q (t h) = q (t)$ and the cocycle law, gives $A_{h} (g t) = A_{h} (t)$ ; Tonelli and the measure-preserving change $(g, t) \mapsto (g t, t)$ show that $A_{h} (u) = A_{h} (t)$ for Haar-a.e. $(u, t)$ , so $A_{h}$ is Haar-a.e. constant, equal to some $σ (h) \in U (K)$ ; hence $b (t h) = b (t) σ (h)$ for Haar-a.e. $t$ .

5.1F2step 4.1

$σ$ is a homomorphism: applying [step 4.1] twice, $b (t) σ (h k) = b (t h k) = b (t) σ (h) σ (k)$ a.e., so $σ (h k) = σ (h) σ (k)$ . It is Borel: integrating the matrix coefficients of the Borel $U (K)$ -valued function $A_{h}$ against a fixed positive probability density on $G$ returns the matrix coefficients of $σ (h)$ (because $A_{h} = σ (h)$ a.e.) and is Borel in $h$ by Tonelli; by [F2], applied to the second-countable group $H$ and the target $U (K)$ , $σ$ is strongly continuous.

6.1F1step 4.1step 5.1

Descent: define $F (x, r) = b (s (x) r) σ (r)^{- 1}$ . For $h \in H$ , $F (x, r h) = b (s (x) r h) σ (r h)^{- 1} = b (s (x) r) σ (h) σ (h)^{- 1} σ (r)^{- 1} = F (x, r)$ up to the a.e. statements of [step 5.1]; hence by [F1] there is a Borel $B : G / H \to U (K)$ with $b (t) = B (q (t)) σ (s (q (t))^{- 1} t)$ for Haar-a.e. $t$ .

7.1F1step 3.1step 6.1

Substituting [step 6.1] into [step 3.1] at the points $t$ and $g t$ gives, for every fixed $g$ , $c (g, q (t)) = B (q (g t)) σ (s (q (g t))^{- 1} g t) σ (s (q (t))^{- 1} t)^{- 1} B (q (t))^{- 1}$ for a.e. $t$ ; using $q (g t) = g q (t)$ and the exact section identity $g s (x) = s (g x) h (g, x)$ this is the displayed formula for a.e. $x$ ; the strict section identity then makes the displayed expression an exact Borel cocycle on all of $G \times G / H$ .

7.2step 4.1step 6.1

Uniqueness of $σ$ : if $(B_{i}, σ_{i})$ both factorize $c$ , set $b_{i} (t) = B_{i} (q (t)) σ_{i} (s (q (t))^{- 1} t)$ ; then $b_{2} (g t)^{- 1} b_{1} (g t) = b_{2} (t)^{- 1} b_{1} (t)$ for every $g$ and a.e. $t$ , so [step 4.1] makes $b_{2}^{- 1} b_{1}$ a constant unitary $T$ , and right- $H$ covariance gives $σ_{2} (h) T = T σ_{1} (h)$ for every $h$ . Since a change of gauge multiplies the lifted factorizations on the left, the class of $σ$ is unchanged.

8.1step 5.1step 6.1step 7.1step 7.2F5∎

Steps 5.1, 6.1, 7.1 and 7.2 give a strongly continuous $σ$ , a Borel $B$ with the displayed factorization, its exact cocycle form, and the uniqueness up to gauge, as claimed.

Remarks

The continuity hypothesis is used only in the upgrade step [3.1]; the a.e. cocycle law and the left-invariance arguments are pure Haar-Tonelli computations. No value of an a.e. class is ever evaluated at a prescribed null coset.

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Unitary equivalence of systems of imprimitivity and of the induced representations

Definition

Two systems of imprimitivity $(U, P)$ on $H$ and $(U^{'}, P^{'})$ on $H^{'}$ for the same Borel $G$ -space $X$ (Systems of imprimitivity for a Borel $G$ -space) are unitarily equivalent when there is a unitary $W : H \to H^{'}$ (Hilbert space) with $W U_{g} W^{- 1} = U_{g}^{'}, W P (E) W^{- 1} = P^{'} (E)$ for every $g \in G$ and every Borel $E \subseteq X$ ; if both are transitive on $X = G / H$ (Transitive systems of imprimitivity and their normalized measure class), such a $W$ is called an equivalence of transitive systems. Two strongly continuous unitary representations $σ, σ^{'}$ of a common group (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) are unitarily equivalent when there is a unitary intertwiner between them.

Well-definedness. For a fixed base $X = G / H$ the relation is the natural isomorphism of pairs (strongly continuous unitary representation, projection-valued measure): it is reflexive with $W = I$ , symmetric with $W^{- 1}$ , and transitive with a composite, because conjugation by a unitary preserves the defining identities; the base isomorphism is suppressed from the notation exactly because transitivity fixes the identification $X = G / H$ . The definition introduces no choice and no new existence assertion.

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An induced representation carries a canonical system of imprimitivity on $G / H$

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ closed, and $σ : H \to U (V)$ a strongly continuous unitary representation on a separable Hilbert space $V$ . Let $Π_{σ} = {Ind}_{H}^{G} σ$ be the induced representation on the covariant completion $H_{σ}$ with rho-measure $μ_{ρ}$ . For a Borel set $E \subseteq G / H$ define $P (E)$ on the covariant model by $(P (E) F) (x) = 1_{E} (x H) F (x)$ . Then $P$ is a projection-valued measure on $G / H$ , $P (E)$ is well defined on the completed space of measurable covariant sections, $Π_{σ} (g) P (E) Π_{σ} (g)^{- 1} = P (g E)$ for all $g \in G$ and Borel $E$ , and $(Π_{σ}, P)$ is a system of imprimitivity on $G / H$ with $P (G / H) = I$ . If $H = G$ the base is one point and $P ({G}) = I$ ; if $H = {e}$ one may normalize so that the system is the multiplication system on $L^{2} (G; V)$ with the left regular action $F \mapsto F (g^{- 1} \cdot)$ .

Facts & Assumptions

Given: AC, the second-countable LCH group $G$ , closed $H \leq G$ , a strongly continuous unitary $σ : H \to U (V)$ on separable $V$ , and the induced representation $Π_{σ}$ on the covariant completion with rho-measure $μ_{ρ}$ .

[F1]

The covariant model consists of (classes of) functions $F : G \to V$ with $F (x h) = σ (h)^{- 1} F (x)$ , compactly supported modulo $H$ , with the norm obtained by integrating the descended pointwise norm against $μ_{ρ}$ ; the dense subspace of continuous covariant sections with compact support modulo $H$ generates the completion, and continuous compactly supported covariant generators are dense (Continuous covariant model and measurable completion, Density of averaged covariant generators, Well-defined induced inner product).

[F2]

The induced action is $(Π_{ρ} (g) F) (x) = D_{g} (x H)^{1 / 2} F (g^{- 1} x)$ with $D_{g} (x H) = ρ (g^{- 1} x) / ρ (x)$ ; it preserves the inner product, satisfies $Π_{ρ} (g_{1}) Π_{ρ} (g_{2}) = Π_{ρ} (g_{1} g_{2})$ , and extends to a unitary on the completion (Unitary cocycle-corrected left action, Unitary induction from a closed subgroup, Continuous quotient translation cocycle).

[F3]

$μ_{ρ}$ is a full-support strongly quasi-invariant Radon measure, so the descended norm integral is a genuine $L^{2}$ integral over the standard Borel $G$ -space $G / H$ ; multiplication by the indicator of a Borel set of finite $μ_{ρ}$ -measure is a bounded self-adjoint idempotent on the completed space, and dominated convergence gives strong countable additivity (Existence of rho-functions and quotient measure classes, Quasi-invariant Radon measure on G/H, Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Bounded borel pvm integral).

[F4]

The induced representation is strongly continuous and the system of imprimitivity axioms require the covariance identity $U_{g} P (E) U_{g}^{- 1} = P (g E)$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Systems of imprimitivity for a Borel $G$ -space).

[F5]

For $H = G$ the quotient is a point and the covariant model is $V$ with the action $σ$ ; for $H = {e}$ the rho-measure may be taken to be Haar measure, covariant functions are unconstrained, and the induced space is $L^{2} (G; V)$ with action $F \mapsto F (g^{- 1} \cdot)$ (Left and right cosets $g H$ and $H g$ of a subgroup, Unitary induction from a closed subgroup, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

[F6]

AC is the standing hypothesis, inherited through the rho-measure and induction suppliers (The Axiom of Choice).

Proof

technique · direct

Given: AC, the group, subgroup, representation $σ$ and the induced model of [F1].

1.1F1F2F3

Identify the covariant completion with the square-integrable measurable covariant sections using the density of the continuous covariant generators in [F1]. Thus a Borel-indicator multiple of a section remains in the completed model. On this measurable model, $(P (E) F) (x) = 1_{E} (x H) F (x)$ is covariant: $(P (E) F) (x h) = 1_{E} (x H) F (x h) = σ (h)^{- 1} (P (E) F) (x)$ , since $x h H = x H$ . It is idempotent and self-adjoint for the induced inner product because $1_{E}^{2} = 1_{E} = \overline{1_{E}}$ pointwise, and it is a contraction: the pointwise norm of $P (E) F$ is at most that of $F$ everywhere. Hence $P (E)$ extends uniquely to a bounded self-adjoint idempotent on $H_{σ}$ .

2.1F1F3step 1.1

$P (\emptyset) = 0$ , $P (G / H) = I$ , and $P (E) P (F) = P (E \cap F)$ follow pointwise from the same identities for indicators, hence hold on the completion by density; strong countable additivity holds because for a disjoint union $E = ⨆ E_{n}$ the partial sums converge pointwise to $1_{E}$ and are bounded, so dominated convergence in the $L^{2}$ -integral gives $P (⋃_{n \leq N} E_{n}) ξ \to P (E) ξ$ for every $ξ$ . Thus $P$ is a projection-valued measure on the Borel $σ$ -algebra of $G / H$ .

2.2F2step 1.1

Covariance: by [F2], $(Π (g) P (E) Π (g)^{- 1} F) (x) = D_{g} (x)^{1 / 2} (P (E) Π (g)^{- 1} F) (g^{- 1} x) = D_{g} (x)^{1 / 2} 1_{E} (g^{- 1} x) (Π (g)^{- 1} F) (g^{- 1} x) = 1_{E} (g^{- 1} x) F (x) = 1_{g E} (x) F (x)$ , so $Π (g) P (E) Π (g)^{- 1} = P (g E)$ for all $g$ and Borel $E$ , first on the dense model and then everywhere by continuity.

3.1F4step 2.1step 2.2

Consequently $(Π_{σ}, P)$ is a system of imprimitivity: $P$ is a PVM by [step 2.1], $Π_{σ}$ is a strongly continuous unitary representation, and the covariance identity is [step 2.2], with $P (G / H) = I$ .

3.2F5step 2.1step 2.2

Boundary cases of the statement: if $H = G$ then $G / H$ is a singleton and the only Borel sets are $\emptyset$ and the point, so $P ({G}) = P (G / H) = I$ and the system is the given representation with the trivial base. If $H = {e}$ then $G / H = G$ , covariant functions are arbitrary, and with the Haar normalization $ρ \equiv 1$ the induced action is $F \mapsto F (g^{- 1} \cdot)$ on $L^{2} (G; V)$ while $P (E)$ is pointwise multiplication by $1_{E}$ ; this is the multiplication system of the statement.

4.1step 2.1step 2.2step 3.1step 3.2F6∎

Steps 2.1, 3.1 and 3.2 prove that $P$ is a well-defined projection-valued measure on the completed space, that the pair $(Π_{σ}, P)$ is a system of imprimitivity on $G / H$ , and the two boundary identifications.

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Spectral multiplicity model of a transitive system of imprimitivity

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ a closed subgroup, and let $(U, P)$ be a transitive system of imprimitivity on $X = G / H$ acting on a nonzero separable Hilbert space $H_{0}$ . Then there exist a finite Borel measure $μ$ on $G / H$ in the quasi-invariant class, a nonzero separable Hilbert space $K$ , and a unitary $W : H_{0} ⟶ L^{2} (G / H, μ; K)$ such that $W P (E) W^{- 1} = M_{1_{E}}$ for every Borel $E \subseteq G / H$ . Moreover $μ$ is quasi-invariant under every $g \in G$ , and the multiplicity is constant almost everywhere; any two such normalizations differ by a decomposable unitary, so $K$ is determined up to isometric isomorphism and $μ$ up to equivalence.

Facts & Assumptions

Given: AC, the transitive system $(U, P)$ on $X = G / H$ with nonzero separable $H_{0}$ .

[F1]

The base $X = G / H$ is a standard Borel space, and the quasi-invariant class is the unique class of nonzero quasi-invariant measures; a Borel set invariant up to null sets for the class is null or conull (Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel, Transitive systems of imprimitivity and their normalized measure class, Quasi-invariant Radon measure on G/H, Existence of rho-functions and quotient measure classes, A transitive Borel $G$ -space with a quasi-invariant measure class is ergodic).

[F2]

Multiplicity model over a standard Borel base: for a PVM $P$ on $X$ and a $P$ -faithful finite Borel measure $μ_{0}$ there are a Borel $m : X \to {1, 2, \dots} \cup {\infty}$ and a unitary $W : H_{0} \to \int_{X}^{\oplus} C^{m (x)} d μ_{0} (x)$ with $W P (E) W^{- 1} = M_{1_{E}}$ for all Borel $E$ ; $P$ -faithful measures exist and any two are mutually absolutely continuous (Multiplicity model of a projection-valued measure over a standard Borel base, Direct integral of a measurable Hilbert field, Measurable Hilbert field from a countable fundamental family, Direct integrals of measurable Hilbert fields are Hilbert spaces).

[F3]

Unitary intertwiners preserve fibre multiplicity: if $U : L^{2} (X, μ; m) \to L^{2} (X, μ; m^{'})$ is unitary with $U M_{f} = M_{f} U$ for all bounded Borel $f$ , then $m = m^{'}$ a.e.; two normalizations of one model over a fixed base therefore differ by a decomposable unitary with unitary fibres a.e. (Unitary intertwiners preserve fibre multiplicity over a standard Borel base).

[F4]

Transport and Radon–Nikodym: for bimeasurable base homeomorphisms and mutually absolutely continuous finite measures there are unitaries of the associated $L^{2}$ -direct-integrals intertwining the multiplication actions, with multiplication by the square root of the appropriate density; the diagonal commutant identifies the intertwining operators as decomposable (Direct integrals transport along bimeasurable base isomorphisms, A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density, Decomposable operators are the commutant of diagonal multiplication).

[F5]

For a Borel $E$ , $μ_{0} (E) = 0 ⟺ P (E) = 0 ⟺ P (g E) = 0 ⟺ μ_{0} (g E) = 0$ , because $P (g E) = U_{g} P (E) U_{g}^{- 1}$ and conjugation by a unitary preserves zero projections; hence any $P$ -faithful $μ_{0}$ is quasi-invariant (Systems of imprimitivity for a Borel $G$ -space, Scalar and complex measures from a pvm, Bounded borel pvm integral).

[F6]

AC is the standing hypothesis (The Axiom of Choice, Separability: the existence of an at most countable dense subset, Hilbert space).

Proof

technique · direct

Given: AC, the transitive system $(U, P)$ on $X = G / H$ .

1.1F1F2F5

Choose a $P$ -faithful finite Borel measure $μ_{0}$ on $X$ by [F2] and apply the multiplicity model: there are a Borel $m : X \to {1, 2, \dots} \cup {\infty}$ and a unitary $W : H_{0} \to \int_{X}^{\oplus} C^{m (x)} d μ_{0} (x)$ with $W P (E) W^{- 1} = M_{1_{E}}$ for every Borel $E$ . By [F5] $μ_{0}$ is quasi-invariant, so $μ_{0}$ lies in the normalized class of $G / H$ .

2.1F5step 1.1

Conjugate the representation: $T_{g} : = W U_{g} W^{- 1}$ is a unitary of the model with $T_{g} M_{f} T_{g}^{- 1} = M_{f \circ g^{- 1}}$ for every bounded Borel $f$ , because $W$ conjugates $P (E)$ to $M_{1_{E}}$ and $U_{g} P (E) U_{g}^{- 1} = P (g E)$ .

3.1F4step 2.1

For each $g$ , form the unitary $Θ_{g} : = c_{g}^{*} T_{g}$ where $c_{g}^{*}$ is the transport unitary associated with the base homeomorphism $x \mapsto g x$ ; here $c_{g}^{*}$ sends $η$ to $x \mapsto η (g x)$ , from the model over $(X, μ_{0}; m)$ to the pulled-back model over $(X, (g^{- 1})_{*} μ_{0}; m \circ g)$ , and intertwines $M_{f \circ g^{- 1}}$ with $M_{f}$ , so $Θ_{g}$ is a unitary $L^{2} (X, μ_{0}; m) \to L^{2} (X, (g^{- 1})_{*} μ_{0}; m \circ g)$ with $Θ_{g} M_{f} = M_{f} Θ_{g}$ . Since $(g^{- 1})_{*} μ_{0}$ is equivalent to $μ_{0}$ by [F5], the Radon–Nikodym isometry $J_{g}$ of [F4] converts it into a unitary $U_{g}^{'} : = J_{g} Θ_{g} : L^{2} (X, μ_{0}; m) \to L^{2} (X, μ_{0}; m \circ g)$ with $U_{g}^{'} M_{f} = M_{f} U_{g}^{'}$ for all bounded Borel $f$ .

4.1F3step 3.1

By the rigidity lemma [F3] applied to $U_{g}^{'}$ , the multiplicities agree: $m = m \circ g$ $μ_{0}$ -almost everywhere, for every $g$ (replacing $g$ by $g^{- 1}$ gives the form stated in the strategy). Therefore each level set ${m = k}$ is invariant under the action up to $μ_{0}$ -null sets.

5.1F1step 4.1

Ergodicity forces one level set to be conull: the countably many level sets partition $X$ , each is invariant up to null sets, so by [F1] each is null or conull; since $μ_{0}$ is nonzero and finite, exactly one level set $X_{0} = {m = k_{0}}$ is conull, and $k_{0} \in {1, 2, \dots} \cup {\infty}$ . Restrict the model to $X_{0}$ : the restriction of $W$ is a unitary $H_{0} \to L^{2} (X_{0}, μ_{0}; k_{0})$ and, viewed on $X$ by zero extension outside $X_{0}$ , a unitary $W_{0} : H_{0} \to L^{2} (X, μ_{0}; K)$ with $K = C^{k_{0}}$ ( $K = ℓ^{2}$ if $k_{0} = \infty$ ), nonzero and separable, and $W_{0} P (E) W_{0}^{- 1} = M_{1_{E}}$ for every Borel $E$ .

6.1F2F3step 5.1

This proves existence with constant multiplicity and quasi-invariant $μ = μ_{0}$ . Uniqueness: if $(W_{1}, μ_{1}, K_{1})$ and $(W_{2}, μ_{2}, K_{2})$ are two such normalizations, $W_{2} W_{1}^{- 1}$ is a unitary intertwining the two multiplication actions over any common base; taking $μ_{1}$ as the base and using mutual absolute continuity, [F3] gives $K_{1} ≅ K_{2}$ isometrically and identifies the intertwiners as decomposable with unitary fibres, while $μ_{1} \sim μ_{2}$ by mutual absolute continuity of $P$ -faithful measures.

7.1step 1.1step 6.1step 4.1step 5.1F6∎

Steps 4.1 and 5.1 establish the model and the constancy of the multiplicity, and step 6.1 gives the stated uniqueness; the measure is quasi-invariant by [step 1.1].

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Measurable cocycle fields for a multiplicity-normalized system

Statement

Assume AC. Let $(U, P)$ be a transitive system of imprimitivity on $G / H$ and fix a normalization $W : H_{0} \to L^{2} (G / H, μ; K)$ with $W P (E) W^{- 1} = M_{1_{E}}$ , $G$ second countable locally compact, $H$ closed, $K$ separable, $μ$ a nonzero $σ$ -finite quasi-invariant Borel measure. For $g \in G$ let $V_{g}$ be the canonical translation operator $(V_{g} f) (x) = [d ((L_{g})_{*} μ) / d μ (x)]^{1 / 2} f (g^{- 1} x)$ and put $W_{g} = V_{g}^{- 1} W U_{g} W^{- 1}$ . Then $W_{g}$ commutes with all multiplications and is therefore multiplication by an essentially bounded measurable operator field $x \mapsto φ_{g} (x) \in B (K)$ ; the field may be chosen so that $(g, x) \mapsto ⟨ φ_{g} (x) ξ, η ⟩$ is Borel on $G \times G / H$ for $ξ, η$ in a fixed dense countable subset of $K$ and so that $φ_{g} (x)$ is unitary for almost every $x$ and every $g$ , with the cocycle identity $φ_{g_{1} g_{2}} (x) = φ_{g_{1}} (g_{2} x) φ_{g_{2}} (x)$ holding for every $g_{1}, g_{2}$ and almost every $x$ .

Facts & Assumptions

Given: AC, the transitive system with its normalization $W$ and the data of the statement.

[F1]

The normalization is unitary with $W P (E) W^{- 1} = M_{1_{E}}$ ; for every bounded Borel $f$ one has $T_{g} M_{f} T_{g}^{- 1} = M_{f \circ g^{- 1}}$ for $T_{g} : = W U_{g} W^{- 1}$ , and $μ$ may be taken to be a finite measure in the quasi-invariant class with $L^{2} (G / H, μ; K)$ the direct integral of the constant field $C^{\dim K}$ (Spectral multiplicity model of a transitive system of imprimitivity, Direct integral of a measurable Hilbert field).

[F2]

The translation operators $V_{g}$ are unitary and $g \mapsto V_{g}$ is strongly continuous on the induced model: $V_{g}$ is the induced action of the trivial representation of $H$ , and the criterion for strong continuity of unitary representations applies (Unitary cocycle-corrected left action, Continuity criteria for unitary representations, Independence of rho and equivalent quotient representative).

[F3]

The commutant of the diagonal multiplications ${M_{f}}$ on a direct integral is exactly the set of decomposable operators, and an essentially bounded weakly measurable field acts decomposably and is unique up to a null set; multiplication by a unitary operator corresponds to a field that is unitary almost everywhere (Decomposable operators are the commutant of diagonal multiplication, Measurable essentially bounded operator fields act decomposably, Measurable and decomposable operator fields, Unitary intertwiners preserve fibre multiplicity over a standard Borel base).

[F4]

The base $G / H$ is standard Borel. Its constant-field direct integral is separable; AC fixes a countable norm-dense sequence with Borel section representatives and a countable orthonormal basis of $K$ . Tonelli applies to sums of squared errors (Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups, Standard Borel spaces, Measurable sections have measurable pointwise inner products, Composition with a Borel measurable outer map preserves measurability, Direct integrals transport along bimeasurable base isomorphisms, Direct integrals of measurable Hilbert fields are Hilbert spaces, A Hilbert space with a dense sequence has a finite or countable orthonormal basis, Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

[F5]

Radon–Nikodym densities of the quasi-invariant measure class are measurable and finite a.e., and the resulting $L^{2}$ -multiplications are measurable in the parameters (A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density, Monotone convergence for the integral).

[F6]

AC is the standing hypothesis (The Axiom of Choice).

Proof

technique · direct

Given: AC, the normalized transitive system and the operators $W_{g}$ .

1.1F1F2F5

If $K = 0$ , choose the sole operator on each fibre; all conclusions are immediate. Assume $K \neq 0$ . By the Haar-lift lemma, $μ$ is equivalent to a rho-derived Radon measure $ν$ . Put $a = d μ / d ν$ , choosing a finite positive Borel version off a null set, and $J f = \sqrt{a} f$ . The set-integral formula shows that $J : L^{2} (μ; K) \to L^{2} (ν; K)$ is unitary and that $d (g_{*} μ) / d μ (x) = a (g^{- 1} x) a (x)^{- 1} d (g_{*} ν) / d ν (x)$ a.e. Hence $J V_{g} J^{- 1} = V_{g}^{ν}$ . The latter is the strongly continuous scalar induced translation tensored with $I_{K}$ , as verified first on finite sums of scalar sections times fibre vectors and then by density. Thus $V_{g}$ is a strongly continuous unitary representation, and so is $T_{g} = W U_{g} W^{- 1}$ ; their product $W_{g} = V_{g}^{- 1} T_{g}$ is strongly continuous and unitary.

2.1F1F3step 1.1algebra

$W_{g}$ commutes with all $M_{f}$ : both $T_{g}$ and $V_{g}$ conjugate $M_{f}$ to $M_{f \circ g^{- 1}}$ , so $V_{g}^{- 1} T_{g} M_{f} = M_{f} V_{g}^{- 1} T_{g}$ . The commutant theorem therefore gives a measurable field representing $W_{g}$ ; the fibrewise identities for $W_{g}^{*} W_{g} = W_{g} W_{g}^{*} = I$ make its fibres unitary almost everywhere.

3.1F3F4F6step 1.1step 2.1construct

Construct a joint representative without changing the fixed measure. Choose a finite-measure Borel partition $(A_{l})$ of $X = G / H$ and put $r (x) = 2^{- l} (1 + μ (A_{l}))^{- 1 / 2}$ on $A_{l}$ ; then $r > 0$ is Borel and belongs to $L^{2} (μ)$ . Fix an orthonormal basis $(e_{j})$ of $K$ and a countable norm-dense sequence of Borel sections $(u_{l})$ by [F4]. For each $j, n$ , choose the least $l = l (g, j, n)$ with $∥ u_{l} - W_{g} (r e_{j}) ∥_{2} < 2^{- n}$ . Its level sets are Borel in $g$ by step 1.1, so $u_{l (g, j, n)} (x)$ is jointly Borel. For each fixed $g, j$ , the sum of squared $L^{2}$ errors is finite. Tonelli, using any representative of $W_{g} (r e_{j})$ , makes the pointwise squared errors summable a.e.; hence the approximants converge a.e. Their limits divided by $r (x)$ are the columns of the field from step 2.1. The set where any column limit fails, or where the columns fail to be a complete orthonormal family, is jointly Borel: convergence, Gram identities, and Parseval on the fixed basis are countably many Borel conditions. Set the field to $I$ there. This gives a jointly Borel $U (K)$ -valued field representing $W_{g}$ for every fixed $g$ .

4.1F3step 1.1step 3.1

For every finite-measure Borel $E$ and basis vector $e_{j}$ , strong continuity of $W_{g}$ gives $\int_{E} ∥ φ_{g} (x) e_{j} - φ_{g_{0}} (x) e_{j} ∥^{2} d μ (x) \to 0$ . Chebyshev's inequality then gives local convergence in measure of each basis column. Finite linear combinations approximate every fibre vector uniformly under unitaries, so this is convergence in measure in the strong topology of $U (K)$ .

4.2F3step 1.1step 2.1step 3.1algebra

Expanding $T_{g_{1} g_{2}} = T_{g_{1}} T_{g_{2}}$ and $V_{g_{1} g_{2}} = V_{g_{1}} V_{g_{2}}$ gives $W_{g_{1} g_{2}} = V_{g_{2}}^{- 1} W_{g_{1}} V_{g_{2}} W_{g_{2}}$ . The first conjugated multiplier has field $x \mapsto φ_{g_{1}} (g_{2} x)$ . Uniqueness of decomposable fields therefore gives $φ_{g_{1} g_{2}} (x) = φ_{g_{1}} (g_{2} x) φ_{g_{2}} (x)$ for every fixed pair and a.e. $x$ .

5.1step 2.1step 3.1step 4.1step 4.2F6∎

Steps 2.1, 3.1, 4.1 and 4.2 give the decomposable unitary fields, a jointly Borel representative, local-measure continuity, and the pairwise a.e. cocycle law, respectively. No representative has been evaluated at a prescribed null coset.

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The stabilizer acts unitarily on an imprimitivity fibre

Statement

Assume AC and let $(U, P)$ be a transitive system on $G / H$ on a separable Hilbert space, with multiplicity-normalized model and source-variable cocycle fields $φ_{g}$ as above. Fix a Borel section $s$ with $s (e H) = e$ . There exist a Borel unitary field $B_{x}$ and a strongly continuous unitary representation $σ : H \to U (K)$ , unique up to unitary equivalence, such that for every $g$ and almost every $x$ , $φ_{g} (x) = B_{g x} σ (h (g, x)) B_{x}^{- 1},$ where $h (g, x) = s (g x)^{- 1} g s (x)$ . The representatives can be replaced by this strict formula on all pairs and normalized with $B_{e H} = I$ , so that $σ (h) = φ_{h} (e H)$ and $B_{x} = φ_{s (x)} (e H)$ . Changes of fields or section give equivalent $σ$ . If $H = G$ one recovers the original representation; if $H$ is trivial the recovered representation is trivial.

Facts & Assumptions

Given: AC, the normalized transitive system, its cocycle fields $φ_{g}$ , a Borel section $s$ with $s (e H) = e$ , and the section cocycle $h (g, x) = s (g x)^{- 1} g s (x)$ .

[F1]

The cocycle fields $φ_{g}$ may be chosen jointly Borel on $G \times G / H$ , unitary for every $g$ and a.e. $x$ , with the a.e. cocycle law and with $g \mapsto φ_{g}$ continuous in local measure in the strong topology; they represent the operators $W_{g} = V_{g}^{- 1} W U_{g} W^{- 1}$ (Measurable cocycle fields for a multiplicity-normalized system).

[F2]

Haar regularization: every such Borel $U (K)$ -valued cocycle factors as $c (g, x) = B (g x) σ (h (g, x)) B (x)^{- 1}$ for a Borel unitary field $B$ and a strongly continuous unitary $σ : H \to U (K)$ , and $σ$ is unique up to unitary equivalence under Borel gauge changes (Haar regularization of transitive unitary cocycles).

[F3]

The section satisfies $s (e H) = e$ , $q \circ s = i d$ , and the section cocycle satisfies the strict identity $h (g_{1} g_{2}, x) = h (g_{1}, g_{2} x) h (g_{2}, x)$ ; moreover $h (h^{'}, e H) = h^{'}$ for $h^{'} \in H$ and $h (s (x), e H) = e$ (Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups, Left and right cosets $g H$ and $H g$ of a subgroup, Left group actions, transitive actions, and faithful actions).

[F4]

Unitary fields over a standard Borel base may be modified on null sets, conjugated pointwise, and evaluated at points after being placed in strict form; changes on null sets do not change the a.e. class of the field, and conjugating the whole factorization by a fixed unitary does not change the equivalence class of $σ$ (Standard Borel spaces, Hilbert space, Separability: the existence of an at most countable dense subset, Hilbert-adjoint identities, Unitary equivalence of systems of imprimitivity and of the induced representations).

[F5]

$U (K)$ with the strong topology is a second-countable topological group, and Borel homomorphisms from the second-countable group $H$ into it are strongly continuous (Steinhaus and Pettis: Borel homomorphisms of second-countable locally compact groups are continuous).

[F6]

AC is the standing hypothesis (The Axiom of Choice, Systems of imprimitivity for a Borel $G$ -space, Transitive systems of imprimitivity and their normalized measure class, Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

Proof

technique · direct

Given: AC, the transitive system with normalized model, the cocycle fields, and the section $s$ .

1.1F1F2

The assignment $c (g, x) : = φ_{g} (x)$ is a Borel $U (K)$ -valued cocycle on $G \times G / H$ satisfying the a.e. cocycle law and the local-measure continuity of [F1]; hence [F2] applies and produces a Borel unitary field $B$ and a strongly continuous unitary $σ : H \to U (K)$ with $φ_{g} (x) = B (g x) σ (h (g, x)) B (x)^{- 1}$ for every $g$ and a.e. $x$ .

1.2F2F4

Normalization at $e H$ : if ${e H}$ is $μ$ -null, redefine $B_{e H} = I$ ; this changes $B$ on a null set and the factorization remains valid a.e. If ${e H}$ is an atom, replace $B_{x}$ by $B_{x} B_{e H}^{- 1}$ and $σ$ by $B_{e H} σ (\cdot) B_{e H}^{- 1}$ , which is a unitary equivalence of representations and makes the new field equal to $I$ at $e H$ . In both cases the factorization holds for every $g$ and a.e. $x$ , and $B_{e H} = I$ .

1.3F2F3

Uniqueness and gauge: if $(B_{1}, σ_{1})$ and $(B_{2}, σ_{2})$ both factorize the same cocycle, the uniqueness clause of [F2] gives a single unitary $T$ with $σ_{2} (h) T = T σ_{1} (h)$ for all $h$ ; a Borel gauge change $B \mapsto A B$ multiplies the lifted trivializations on the left and does not change the equivalence class. A change of section changes $B$ by the corresponding $σ$ -factor and leaves the class of $σ$ fixed.

2.1F3step 1.1

Place the formula in strict form: define ${\hat{φ}}_{g} (x) : = B (g x) σ (h (g, x)) B (x)^{- 1}$ ; by [step 1.1] ${\hat{φ}}_{g} = φ_{g}$ a.e. for every $g$ , and the right-hand side is jointly Borel in $(g, x)$ ; the strict section identity of [F3] makes $\hat{φ}$ an exact cocycle on all of $G \times G / H$ , so replacing the original fields by $\hat{φ}$ changes nothing in the a.e. class and gives the displayed formula for every pair.

3.1F3step 1.2step 2.1

Evaluating the strict formula at $x = e H$ : for $h^{'} \in H$ one has $h (h^{'}, e H) = s (e H)^{- 1} h^{'} s (e H) = h^{'}$ , so $φ_{h^{'}} (e H) = B (e H) σ (h^{'}) B (e H)^{- 1} = σ (h^{'})$ because $B_{e H} = I$ ; and for $g = s (x)$ , $h (s (x), e H) = s (x)^{- 1} s (x) s (e H) = e$ gives $φ_{s (x)} (e H) = B (x) σ (e) B_{e H}^{- 1} = B (x)$ . Thus the recovered data are exactly $σ (h^{'}) = φ_{h^{'}} (e H)$ and $B_{x} = φ_{s (x)} (e H)$ .

4.1F3F4step 3.1

Boundary cases: if $H = G$ then $G / H$ is a point, $s (e H) = e$ , and the factorization collapses to $φ_{g} = B σ (g) B^{- 1}$ , so $σ$ is unitarily equivalent to the original representation carried by the fields. If $H = {e}$ then $H$ is the trivial group and $σ$ is a strongly continuous unitary representation of the trivial group, hence the identity representation on its given fibre $K$ ; nothing more is asserted.

5.1step 1.1step 2.1step 3.1step 1.3step 4.1F5F6∎

Steps 1.1, 2.1 and 3.1 give existence of $B, σ$ with the strict factorization and the two evaluation identities; [step 1.3] gives uniqueness up to unitary equivalence and gauge; [step 4.1] gives the two boundary cases. This proves the statement.

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The imprimitivity reconstruction map is isometric and intertwining

Statement

Assume AC. In the normalized model of a transitive system $(U, P)$ on $G / H$ , take the Borel unitaries $B$ and the stabilizer representation $σ$ supplied by the preceding lemma. Multiplication by $B$ is unitary and $(B^{- 1} W U_{g} W^{- 1} B f) (x) = D_{g} (x)^{1 / 2} σ (s (x)^{- 1} g s (g^{- 1} x)) f (g^{- 1} x) .$ This is the canonical induced action of ${Ind}_{H}^{G} σ$ . Moreover $B^{- 1} W P (E) W^{- 1} B = M_{1_{E}}$ for all Borel $E$ . Hence $(U, P)$ is unitarily equivalent to the canonical induced system by an isometric map intertwining both $U$ and $P$ .

Facts & Assumptions

Given: AC, the normalized transitive system $(U, P)$ with multiplicity model $W$ , cocycle fields $φ_{g}$ , Borel unitaries $B$ and stabilizer representation $σ$ .

[F1]

The multiplicity-normalized model is $W : H_{0} \to L^{2} (G / H, μ; K)$ with $W P (E) W^{- 1} = M_{1_{E}}$ , and the source-variable cocycle fields satisfy $W U_{g} W^{- 1} f (x) = D_{g} (x)^{1 / 2} φ_{g} (g^{- 1} x) f (g^{- 1} x)$ , where $D_{g} (x) = d (L_{g})_{*} μ / d μ (x)$ (Spectral multiplicity model of a transitive system of imprimitivity, Measurable cocycle fields for a multiplicity-normalized system, Direct integral of a measurable Hilbert field).

[F2]

The stabilizer lemma supplies, after the strict normalization, the identity $φ_{g} (x) = B (g x) σ (h (g, x)) B (x)^{- 1}$ for every $g$ and every $x$ , with $h (g, x) = s (g x)^{- 1} g s (x)$ (The stabilizer acts unitarily on an imprimitivity fibre, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

[F3]

The canonical induced model of $σ$ on the covariant completion with rho-measure $μ_{ρ}$ has action $(Π (g) F) (x) = D_{g} (x)^{1 / 2} F (g^{- 1} x)$ on covariant $F$ and is independent of the choice of rho-function and of the equivalent measure representative in the class (An induced representation carries a canonical system of imprimitivity on $G / H$ , Continuous covariant model and measurable completion, Unitary cocycle-corrected left action, Unitary induction from a closed subgroup, Independence of rho and equivalent quotient representative).

[F4]

Multiplication by a Borel field of unitary operators is unitary on the direct integral and commutes with every $M_{f}$ ; the commutant of the multiplications consists of the decomposable operators (Decomposable operators are the commutant of diagonal multiplication, Direct integral of a measurable Hilbert field).

[F5]

AC is the standing hypothesis (The Axiom of Choice, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Transitive systems of imprimitivity and their normalized measure class, Unitary equivalence of systems of imprimitivity and of the induced representations).

Proof

technique · direct

Given: AC, the normalized model with $B, σ$ and the cocycle fields.

1.1F4

Multiplication $M_{B}$ by the Borel unitary field $x \mapsto B_{x}$ is a unitary of $L^{2} (G / H, μ; K)$ by [F4], and it commutes with every $M_{f}$ because $f (x) I_{K}$ commutes with the operator $B_{x}$ in every fibre.

2.1F1F2F3step 1.1

Action computation: for $f$ in the model, using [F1] and then the strict factorization [F2] evaluated at the source point $g^{- 1} x$ , where $h (g, g^{- 1} x) = s (x)^{- 1} g s (g^{- 1} x)$ , $(M_{B}^{- 1} W U_{g} W^{- 1} M_{B} f) (x) = B_{x}^{- 1} D_{g} (x)^{1 / 2} (B_{x} σ (h (g, g^{- 1} x)) B_{g^{- 1} x}^{- 1}) B_{g^{- 1} x} f (g^{- 1} x),$ so the $B$ -factors cancel and the result is $D_{g} (x)^{1 / 2} σ (s (x)^{- 1} g s (g^{- 1} x)) f (g^{- 1} x)$ . By [F3] this is exactly the canonical induced action of ${Ind}_{H}^{G} σ$ in section coordinates: identifying a square-integrable section $f$ with the covariant function determined by $F (s (x)) = f (x)$ and $F (x h) = σ (h)^{- 1} F (x)$ , one has $F (g^{- 1} s (x)) = σ (h (g, g^{- 1} x)) F (s (g^{- 1} x))$ because $g^{- 1} s (x) = s (g^{- 1} x) h (g^{- 1}, x)$ and $h (g^{- 1}, x) = h (g, g^{- 1} x)^{- 1}$ , so the induced formula $(Π (g) F) (x) = D_{g} (x)^{1 / 2} F (g^{- 1} x)$ becomes the displayed action; the measure $μ$ lies in the class used by the induced model by [F3].

2.2F4step 1.1

PVM transport: $M_{B}$ commutes with $M_{1_{E}}$ , so $M_{B}^{- 1} W P (E) W^{- 1} M_{B} = M_{B}^{- 1} M_{1_{E}} M_{B} = M_{1_{E}}$ for every Borel $E$ .

3.1step 2.1step 2.2

Consequently the composite $Φ : = M_{B}^{- 1} W : H_{0} \to L^{2} (G / H, μ; K)$ is a unitary (a composite of unitaries), and by [step 2.1] and [step 2.2] it intertwines $U_{g}$ with the canonical induced action and $P (E)$ with multiplication by $1_{E}$ . Being unitary, $Φ$ is isometric; the target is identified with the induced space of $σ$ by [F3].

4.1step 3.1F5∎

Thus the transitive system $(U, P)$ is unitarily equivalent to the canonical induced system of $σ$ by the isometric intertwiner $Φ$ , which is the reconstruction map of the statement.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Mackey's imprimitivity theorem

Statement

Assume AC. Let $G$ be a second-countable locally compact Hausdorff topological group, $H \leq G$ a closed subgroup, and $(U, P)$ a transitive system of imprimitivity on $G / H$ acting on a separable Hilbert space $H_{0}$ . Then there exist a strongly continuous unitary representation $σ : H \to U (K)$ on a separable Hilbert space $K$ and a unitary $W : H_{0} ⟶ L^{2} (G / H, μ; K)$ onto the induced space of $σ$ such that $W U_{g} W^{- 1} = {Ind}_{H}^{G} σ (g) (g \in G), W P (E) W^{- 1} = M_{1_{E}} (E \subseteq G / H Borel),$ where $M_{1_{E}}$ is multiplication by the indicator of $E$ on the covariant model. Conversely, for every strongly continuous unitary $σ : H \to U (K)$ on a separable Hilbert space $K$ , the induced representation together with multiplication by indicators on $G / H$ is a transitive system of imprimitivity, and the two constructions are inverse up to unitary equivalence. The uniqueness theorem records the corresponding bijection of equivalence classes.

Facts & Assumptions

Given: AC, the transitive system $(U, P)$ on $G / H$ with separable $H_{0}$ , and the induced-system construction of An induced representation carries a canonical system of imprimitivity on $G / H$ .

[F1]

The multiplicity model of a transitive system provides a finite quasi-invariant measure $μ$ in the normalized class, a separable nonzero $K$ , a Borel multiplicity $m$ constant a.e., and a unitary $W : H_{0} \to L^{2} (G / H, μ; K)$ with $W P (E) W^{- 1} = M_{1_{E}}$ (Spectral multiplicity model of a transitive system of imprimitivity, Transitive systems of imprimitivity and their normalized measure class, Direct integral of a measurable Hilbert field).

[F2]

For $T_{g} = W U_{g} W^{- 1}$ and the canonical scalar translation $V_{g}$ , the operators $W_{g} = V_{g}^{- 1} T_{g}$ are multiplication by jointly Borel, a.e. unitary fields $φ_{g}$ , with $φ_{g_{1} g_{2}} (x) = φ_{g_{1}} (g_{2} x) φ_{g_{2}} (x)$ for each pair and almost every $x$ (Measurable cocycle fields for a multiplicity-normalized system).

[F3]

If the fields of [F2] are continuous in local measure in the strong topology, Haar regularization gives a Borel unitary field $B$ and a strongly continuous unitary $σ : H \to U (K)$ with $φ_{g} (x) = B (g x) σ (s (g x)^{- 1} g s (x)) B (x)^{- 1}$ for every $g$ and almost every $x$ . The formula defines a strict Borel cocycle, and $σ$ is unique up to unitary equivalence (Haar regularization of transitive unitary cocycles, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

[F4]

The reconstruction map $Φ = M_{B}^{- 1} W$ is a unitary onto the canonical induced space of $σ$ intertwining $U$ with ${Ind}_{H}^{G} σ$ and $P$ with multiplication by indicators (The imprimitivity reconstruction map is isometric and intertwining, Continuous covariant model and measurable completion, Unitary induction from a closed subgroup).

[F5]

Conversely, the induced representation ${Ind}_{H}^{G} σ$ together with $P (E) = M_{1_{E}}$ is a transitive system of imprimitivity on $G / H$ with the same normalization, and for $H = G$ (one-point base) and $H = {e}$ (multiplication system) the boundary clauses hold (An induced representation carries a canonical system of imprimitivity on $G / H$ ).

[F6]

Integrating a system of imprimitivity gives a nondegenerate representation of the transformation algebra, so the choice of PVM is not an extra datum once the system is fixed; the zero Hilbert space carries the zero system (A system of imprimitivity integrates to a nondegenerate representation of the transformation algebra, Systems of imprimitivity for a Borel $G$ -space).

[F7]

Unitary equivalence of systems and of representations is the relation of Unitary equivalence of systems of imprimitivity and of the induced representations, and AC is the standing hypothesis (The Axiom of Choice, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Hilbert space, Separability: the existence of an at most countable dense subset).

[F8]

The finite quasi-invariant $μ$ is equivalent to a rho-derived Radon measure $ν$ ; a positive finite Borel version of $a = d μ / d ν$ exists, and scalar unitary induction is strongly continuous. Borel homomorphisms $H \to U (K)$ are strongly continuous when $K$ is separable (Haar null classes and Borel descent on a homogeneous space, A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density, Unitary induction from a closed subgroup, Steinhaus and Pettis: Borel homomorphisms of second-countable locally compact groups are continuous).

Proof

technique · direct

Given: AC, the transitive system $(U, P)$ on $G / H$ and a strongly continuous unitary $σ : H \to U (K)$ on separable $K$ in the converse direction.

1.1F5F6

Zero case: if $H_{0} = {0}$ , take $K = {0}$ , the zero representation of $H$ and the zero unitary; the induced space is ${0}$ , the canonical system has $P (G / H) = I = 0$ , and the displayed identities hold; the same data give the zero system from the zero representation. Hence assume $H_{0} \neq {0}$ .

1.2F1F2F8given

Apply [F1] to obtain $W, μ, K$ and put $T_{g} = W U_{g} W^{- 1}$ ; [F2] gives the Borel source-variable cocycle fields. Choose $ν$ and $a = d μ / d ν$ from [F8] and set $J f = \sqrt{a} f$ . Then $J : L^{2} (μ; K) \to L^{2} (ν; K)$ is unitary. Pushing the equality $d μ = a d ν$ through the base translation gives $D_{g}^{μ} (x) = a (g^{- 1} x) a (x)^{- 1} D_{g}^{ν} (x)$ almost everywhere, so $J V_{g}^{μ} J^{- 1} = V_{g}^{ν}$ . The scalar $V_{g}^{ν}$ is induction of the trivial representation of $H$ and is strongly continuous by [F8]; this extends to $K$ -valued sections first on finite sums $f (x) ξ$ and then by their density and unitarity. Hence $V_{g}^{μ}$ is strongly continuous. Since $T_{g}$ is strongly continuous, so is $W_{g} = (V_{g}^{μ})^{- 1} T_{g}$ , by the triangle inequality and unitary norm bounds.

1.3F5

Reverse direction: given $σ : H \to U (K)$ on separable $K$ , [F5] endows the induced representation on its covariant completion with the multiplication PVM $P (E) = M_{1_{E}}$ , which is a projection-valued measure with $P (G / H) = I$ and the covariance identity, hence a transitive system of imprimitivity on $G / H$ ; this is the converse construction.

2.1F2F7step 1.2algebra

Fix $g_{0} \in G$ , a Borel $E \subseteq G / H$ with $μ (E) < \infty$ , and $ξ \in K$ . Since $1_{E} ξ \in L^{2} (μ; K)$ and $W_{g} = M_{φ_{g}}$ , step 1.2 gives $\int_{E} ∥ (φ_{g} (x) - φ_{g_{0}} (x)) ξ ∥^{2} d μ (x) = ∥ (W_{g} - W_{g_{0}}) (1_{E} ξ) ∥_{2}^{2} \to 0$ . Therefore $μ {x \in E : ∥ (φ_{g} (x) - φ_{g_{0}} (x)) ξ ∥ > ε} \leq ε^{- 2} ∥ (W_{g} - W_{g_{0}}) (1_{E} ξ) ∥_{2}^{2} \to 0$ . On the unitary group, the strong topology is determined by a countable dense set of vectors in separable $K$ : finite-vector tests pass to every vector using $∥ (u - v) (ξ - η) ∥ \leq 2 ∥ ξ - η ∥$ . Finite unions of the displayed exceptional sets thus prove local convergence in measure in the strong topology. This is the continuity hypothesis required in [F3].

3.1F3F8step 2.1

Now [F3] applies with its continuity hypothesis verified by step 2.1 and supplies $B, σ$ . In the Haar proof the lifted coboundary $b$ has $b (t h) = b (t) σ (h)$ for each $h$ and almost every $t$ . Thus $σ (h k) = σ (h) σ (k)$ ; integrating the Borel matrix coefficients of $b (t)^{- 1} b (t h)$ against a fixed Haar probability density makes every coefficient of $σ$ Borel. Since $K$ is separable, [F8] applies to this Borel homomorphism and gives strong continuity. Invariant Borel descent of $b (s (x) h) σ (h)^{- 1}$ gives $B (x)$ , and the resulting section-cocycle formula is strict on all pairs after replacing the fields by their equal a.e. representatives.

4.1F1F2F4step 3.1

Multiplication by the Borel unitaries $B (x)$ is unitary and commutes with indicator multiplications. Put $Φ = M_{B}^{- 1} W$ . Substituting the factorization from step 3.1 at the source point $g^{- 1} x$ gives $(Φ U_{g} Φ^{- 1} f) (x) = D_{g}^{μ} (x)^{1 / 2} σ (s (x)^{- 1} g s (g^{- 1} x)) f (g^{- 1} x)$ and $Φ P (E) Φ^{- 1} = M_{1_{E}}$ . This is the section-coordinate form of the canonical induced action, as in [F4], so $Φ$ is the required unitary and may be denoted $W$ in the statement.

5.1F3F5step 4.1

Start with the canonical system of $σ$ . Its source-variable cocycle is $φ_{g} (x) = σ (s (g x)^{- 1} g s (x))$ , so $(B, σ) = (I, σ)$ is already a factorization. Any recovered representation $σ^{'}$ is unitarily equivalent to $σ$ by the uniqueness clause of [F3]. Conversely step 4.1 reconstructs a system unitarily equivalent to the starting $(U, P)$ . Thus the constructions are inverse up to unitary equivalence.

6.1step 1.1step 1.2step 2.1step 3.1step 4.1step 1.3step 5.1F7∎

Steps 1.2, 2.1, 3.1 and 4.1 prove the forward direction, including the local-measure continuity and strongly continuous stabilizer action; steps 1.3 and 5.1 give the converse and inverse character up to unitary equivalence. The zero case is covered by step 1.1.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Uniqueness in the imprimitivity theorem

Statement

Assume AC and keep the hypotheses of the imprimitivity theorem. If $σ : H \to U (K)$ and $σ^{'} : H \to U (K^{'})$ are strongly continuous unitary representations, then the canonical transitive systems of ${Ind}_{H}^{G} σ$ and ${Ind}_{H}^{G} σ^{'}$ on $G / H$ are unitarily equivalent if and only if $σ$ and $σ^{'}$ are unitarily equivalent. Consequently the map of the imprimitivity theorem is a bijection between unitary equivalence classes of transitive systems on $G / H$ and unitary equivalence classes of strongly continuous unitary representations of $H$ .

Facts & Assumptions

Given: AC, the second-countable LCH group $G$ and closed subgroup $H$ , and strongly continuous unitary representations $σ : H \to U (K)$ , $σ^{'} : H \to U (K^{'})$ on separable spaces.

[F1]

The canonical system of $σ$ is the induced representation on its covariant completion together with the multiplication PVM $P (E) = M_{1_{E}}$ ; the induced action in section coordinates is $D_{g} (x)^{1 / 2} σ (s (x)^{- 1} g s (g^{- 1} x)) f (g^{- 1} x)$ (An induced representation carries a canonical system of imprimitivity on $G / H$ , The imprimitivity reconstruction map is isometric and intertwining, Unitary induction from a closed subgroup).

[F2]

A system equivalence between two multiplicity-normalized models intertwines the diagonal multiplications, so it is decomposable with unitary fibres almost everywhere, and the fibre dimensions agree a.e.; equivalently, over a fixed base the unitary intertwiners of two models are precisely the decomposable unitaries (Unitary intertwiners preserve fibre multiplicity over a standard Borel base, Decomposable operators are the commutant of diagonal multiplication, Spectral multiplicity model of a transitive system of imprimitivity).

[F3]

The cocycle fields of the canonical model of $σ$ factor through a trivialization $b_{σ}$ : writing $φ_{g}^{σ} (x) = σ (s (g x)^{- 1} g s (x))$ at the source variable, the Haar regularization uniqueness argument shows that if two trivializations of the same cocycle differ by a gauge $A$ , then $b_{σ^{'}} (t)^{- 1} A (q (t)) b_{σ} (t)$ is left-translation invariant for a.e. $t$ , hence a constant unitary $T$ , and right- $H$ covariance gives $σ^{'} (h) T = T σ (h)$ for all $h$ (Haar regularization of transitive unitary cocycles, Measurable cocycle fields for a multiplicity-normalized system, The stabilizer acts unitarily on an imprimitivity fibre).

[F4]

The imprimitivity theorem gives the forward and inverse constructions and the zero cases: zero fibres induce exactly the zero system, and a nonzero fibre induces a nonzero space because the quotient measure has full support and nonzero square-integrable sections exist (Mackey's imprimitivity theorem, Transitive systems of imprimitivity and their normalized measure class, Unitary equivalence of systems of imprimitivity and of the induced representations).

[F5]

AC is the standing hypothesis (The Axiom of Choice, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

Proof

technique · direct

Given: AC, the two representations $σ, σ^{'}$ and their canonical systems.

1.1F1

If $σ$ and $σ^{'}$ are unitarily equivalent via $T : K \to K^{'}$ , define $\hat{T}$ on the covariant completion of $Ind σ$ pointwise, $(\hat{T} F) (x) = T (F (x))$ . Then $\hat{T}$ is unitary, preserves covariance ( $T (F (x h)) = T (σ (h)^{- 1} F (x)) = σ^{'} (h)^{- 1} (T F) (x)$ ), and intertwines the induced actions and the multiplication PVM: $\hat{T} Π_{σ} (g) {\hat{T}}^{- 1} = Π_{σ^{'}} (g)$ and $\hat{T} P (E) {\hat{T}}^{- 1} = P^{'} (E)$ . Hence the canonical systems are unitarily equivalent.

2.1F2F3step 1.1

Conversely, suppose the canonical systems are unitarily equivalent by $W_{0}$ . Then $W_{0}$ intertwines all multiplications by indicators, and by [F2] it is multiplication by a Borel unitary field $A (x)$ between the two constant fibres, whose dimensions agree. Fix unitary identifications of the fibres and use [F3]: the two cocycle fields of the canonical models are related by the gauge $A$ , and lifting the gauge to $G$ produces a constant unitary $T$ with $σ^{'} (h) T = T σ (h)$ for every $h \in H$ . Thus $σ$ and $σ^{'}$ are unitarily equivalent.

3.1F4step 2.1

Zero cases: if $K = 0$ then the canonical system is the zero system and $H$ acts trivially; two zero systems are unitarily equivalent, and the zero representation of $H$ is unitarily equivalent only to the zero representation; if both $K, K^{'}$ are nonzero the argument [step 2.1] applies verbatim, and a nonzero fibre induces a nonzero system by [F4], so the zero and nonzero classes do not mix.

4.1step 1.1step 2.1step 3.1F5∎

Steps [1.1], [2.1] and [3.1] show that the canonical construction induces a well-defined bijection between unitary equivalence classes of strongly continuous unitary representations of $H$ and unitary equivalence classes of transitive systems on $G / H$ , in both directions; the map of the imprimitivity theorem is that bijection.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Mackey little-group reduction for an abelian normal subgroup

Statement

Assume AC. Let $G = N ⋊ K$ be a topological semidirect product with continuous automorphism action and product topology, $N$ abelian and closed normal, $G$ second countable locally compact, and let the dual action of $K$ on $\hat{N}$ have regular orbits in the sense of the preceding lemma (equivalently, the orbit space $\hat{N} / K$ is countably separated). For $χ \in \hat{N}$ let $K_{χ} = {k \in K : k \cdot χ = χ}$ and $H_{χ} = N ⋊ K_{χ}$ . Then every irreducible strongly continuous unitary representation $π$ of $G$ is unitarily equivalent to ${Ind}_{H_{χ}}^{G} (χ \otimes θ)$ for some $χ \in \hat{N}$ and some irreducible strongly continuous unitary representation $θ$ of $K_{χ}$ , where $χ \otimes θ$ denotes the representation $(n, k_{χ}) \mapsto χ (n) θ (k_{χ})$ of $H_{χ}$ .

Facts & Assumptions

Given: AC, the semidirect product $G = N ⋊ K$ with abelian closed normal $N$ , an irreducible strongly continuous unitary representation $π$ of $G$ on a separable Hilbert space, and the regular-orbit hypothesis on the dual action.

[F1]

Restricting $π$ to $N$ and letting $K$ act through $π ∣_{K}$ satisfies the covariance hypothesis of the spectral lemma: there is a unique regular PVM $P$ on $\hat{N}$ with $π (n) = \int χ (n) d P (χ)$ and $π (k) P (E) π (k)^{- 1} = P (k \cdot E)$ for all $k$ , with the dual action $k \cdot χ = χ \circ α_{k}^{- 1}$ (Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity, The external semidirect product $N ⋊_{α} H$ , The Pontryagin dual with the compact-open topology, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Left group actions, transitive actions, and faithful actions).

[F2]

If $π$ is irreducible, the system is ergodic: an invariant spectral projection $P (E)$ commutes with $π (N)$ and is invariant under $π (K)$ , hence carries an invariant closed subspace; irreducibility forces it to be $0$ or $I$ (Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity, Schur lemma for complex unitary representations).

[F3]

Under the regular-orbit hypothesis, an ergodic system of imprimitivity on $\hat{N}$ concentrates on a single orbit: there is $χ \in \hat{N}$ with $P (K \cdot χ) = I$ , and the orbit is Borel (Ergodic systems with regular orbits concentrate on one orbit).

[F4]

The dual is second countable and standard Borel by the spectral lemma’s proof step 1.1. The dual action is jointly continuous: for a compact $C \subseteq N$ and a compact neighbourhood $V$ in $K$ , the images $α_{k}^{- 1} (C)$ , $k \in V$ , lie in one compact set; uniform convergence of characters there and continuity of the action give compact-open continuity. For $χ \in \hat{N}$ the stabilizer $K_{χ}$ is closed, the orbit map $K / K_{χ} \to K \cdot χ$ is a continuous bijection onto the Borel orbit, and $G / H_{χ} ≅ K / K_{χ}$ ; these homogeneous spaces are Polish standard Borel with Borel actions (Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity, The dual of a locally compact abelian group is locally compact abelian, Continuity of a map of topological spaces at a point and globally, Left and right cosets $g H$ and $H g$ of a subgroup, Compact lifts and averaging onto C_c(G/H), Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel).

[F5]

Mackey's imprimitivity theorem applies to the transported transitive system: there are a strongly continuous unitary $σ : H_{χ} \to U (K_{0})$ and a unitary intertwining $π$ with ${Ind}_{H_{χ}}^{G} σ$ (Mackey's imprimitivity theorem, Transitive systems of imprimitivity and their normalized measure class, Unitary equivalence of systems of imprimitivity and of the induced representations).

[F6]

In the normalized induced model of the system concentrated on $K \cdot χ$ , the action of $N$ is multiplication by the character $χ$ evaluated at the source point, the gauge commutes with the scalar action of $N$ , and the induced formula for $σ$ gives $σ (s (x)^{- 1} n s (x)) = (s (x) \cdot χ) (n) I$ for a.e. $x$ ; since $N$ is normal, conjugation by $s (x)$ maps $N$ onto itself, and strong continuity extends the identity from a countable dense subset of $N$ to all of $N$ (Haar null classes and Borel descent on a homogeneous space, An induced representation carries a canonical system of imprimitivity on $G / H$ , Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups, Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F7]

AC is the standing hypothesis (The Axiom of Choice, Systems of imprimitivity for a Borel $G$ -space).

Proof

technique · direct

Given: AC, the semidirect product, the irreducible $π$ , and the regular-orbit hypothesis.

1.1F1F2F3F4

The representation space is separable even if this was not assumed. For $v \neq 0$ , the closed span of $π (G) v$ is invariant and hence is the whole space by irreducibility. A countable dense subset $D \subseteq G$ exists because $G$ is second-countable LCH; strong continuity makes $π (D) v$ dense in the orbit. Its finite rational-complex linear combinations are countable and dense in the Hilbert space. Thus [F1] applies. It produces $P$ , which is ergodic by [F2] and concentrates on a Borel orbit $C = K \cdot χ$ by [F3].

2.1F3F4F5step 1.1

The continuous orbit bijection $r : K / K_{χ} \to C$ of [F4] is bimeasurable. Indeed, every open subset $O$ of the second-countable LCH quotient is a countable union of compact sets contained in $O$ : use a countable base with compact closures and shrink inside $O$ . Their images under $r$ are compact, hence closed in the Hausdorff dual, so $r (O)$ is Borel. This proves measurability of $r^{- 1}$ without assuming that $r$ is a homeomorphism. Transporting $P ∣_{C}$ gives a transitive system for $G$ on $G / H_{χ} ≅ K / K_{χ}$ ; $N$ acts trivially on the base. By [F5], $π ≅ {Ind}_{H_{χ}}^{G} σ$ for a strongly continuous $σ$ .

3.1F4F6step 2.1

Choose the Borel section in $G$ with values $s (x) \in K$ . For every $n \in N$ , the spectral formula makes $π (n)$ multiplication by $x (n)$ on the orbit, and the reconstruction gauge commutes with this scalar multiplier. Since $N$ fixes the base, the induced Radon–Nikodym factor is one and the induced formula gives $σ (s (x)^{- 1} n s (x)) = χ (s (x)^{- 1} n s (x)) I$ for a.e. $x$ . Intersect these conull sets over a countable dense subset of $N$ and fix one $x$ in the intersection. Continuity of both sides extends the identity to all $n \in N$ at this $x$ . Conjugation by $s (x)$ maps $N$ onto itself, so $σ (m) = χ (m) I$ for all $m \in N$ . Put $θ = σ ∣_{K_{χ}}$ ; it is strongly continuous and $σ (n, k) = χ (n) θ (k)$ . Stabilizer invariance of $χ$ verifies multiplicativity of this formula in the semidirect product.

4.1F5step 3.1

$θ$ is irreducible: if $θ$ had a nontrivial closed invariant subspace, inducing it would produce a nontrivial closed invariant subspace of ${Ind}_{H_{χ}}^{G} (χ \otimes θ) ≅ π$ , because the quotient measure class has full support so a nonzero fibrewise subspace induces a nonzero closed subspace; this contradicts irreducibility of $π$ .

5.1F5step 1.1step 3.1step 4.1F7∎

Therefore $π$ is unitarily equivalent to ${Ind}_{H_{χ}}^{G} (χ \otimes θ)$ with $χ \in \hat{N}$ and $θ$ an irreducible strongly continuous unitary representation of $K_{χ}$ , as claimed; the orbit $χ$ is the one selected by the spectral PVM, and the inducing class is determined by the system uniqueness theorem.

5 · Examples, counterexamples and false statements

None yet.

Mackeys Imprimitivity Theorem

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Systems of imprimitivity for a Borel GG-space

Definition

The transformation (covariance) algebra Cc(G×X)Cc​(G×X)

Definition

Characters of the L1 algebra of an abelian group

Statement

Facts & Assumptions

Proof

Remarks

Direct integrals transport along bimeasurable base isomorphisms

Statement

Facts & Assumptions

Proof

Nondegenerate representations of C0 have regular PVMs

Statement

Facts & Assumptions

Proof

Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel

Statement

Facts & Assumptions

Proof

Steinhaus and Pettis: Borel homomorphisms of second-countable locally compact groups are continuous

Statement

Facts & Assumptions

Proof

Remarks

Unitary intertwiners preserve fibre multiplicity over a standard Borel base

Statement

Facts & Assumptions

Proof

A system of imprimitivity integrates to a nondegenerate representation of the transformation algebra

Statement

Facts & Assumptions

Proof

Remarks

Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups

Statement

Facts & Assumptions

Proof

Remarks

LCA Fourier transforms form a dense algebra in C0 of the dual

Statement

Facts & Assumptions

Proof

Multiplicity model of a projection-valued measure over a standard Borel base

Statement

Facts & Assumptions

Proof

Ergodic systems with regular orbits concentrate on one orbit

Statement

Facts & Assumptions

Proof

Haar null classes and Borel descent on a homogeneous space

Statement

Facts & Assumptions

Proof

Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity

Statement

Facts & Assumptions

Proof

Transitive systems of imprimitivity and their normalized measure class

Definition

A transitive Borel GG-space with a quasi-invariant measure class is ergodic

Statement

Facts & Assumptions

Proof

Haar regularization of transitive unitary cocycles

Statement

Facts & Assumptions

Proof

Remarks

Unitary equivalence of systems of imprimitivity and of the induced representations

Definition

An induced representation carries a canonical system of imprimitivity on G/HG/H

Statement

Systems of imprimitivity for a Borel $G$ -space

The transformation (covariance) algebra $C_{c} (G \times X)$

A transitive Borel $G$ -space with a quasi-invariant measure class is ergodic

An induced representation carries a canonical system of imprimitivity on $G / H$