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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.

Depends on

Used by

The stabilizer acts unitarily on an imprimitivity fibre Lemma
Mackey's imprimitivity theorem Theorem
Uniqueness in the imprimitivity theorem Theorem

Dependency tree · two levels

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Sources

G. W. Mackey, Imprimitivity for Representations of Locally Compact Groups I, PNAS 35 (1949) 537-545 (Internet Archive capture of the PubMed Central scan) (standard reference, not scraped)