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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.

Depends on

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Sources

V. S. Sunder, Notes on the Imprimitivity Theorem (ISIBangalore/IMSc lecture notes, 22 pp.) (standard reference, not scraped)
G. Misra, E. K. Narayanan and C. Varughese, Mackey Imprimitivity and commuting tuples of homogeneous normal operators, arXiv:2402.15737 (standard reference, not scraped)
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)