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Essential uniqueness of the central decomposition

Statement

Assume the Axiom of Choice. Let $G$ be a second-countable locally compact group and $(π, H)$ a separable strongly continuous unitary representation with central decompositions over standard Borel spaces $(X, μ)$ and $(Y, ν)$ as in Central decomposition into factor representations. Then the decompositions agree up to a bimeasurable base isomorphism and a null-set modification: there are conull Borel sets $X_{0}, Y_{0}$ and a bimeasurable bijection $c : X_{0} \to Y_{0}$ with $c_{*} (μ ∣_{X_{0}})$ equivalent to $ν ∣_{Y_{0}}$ such that the fibre factor representations are unitarily equivalent almost everywhere via a measurable field $x \mapsto u_{x}$ : $u_{x} π_{x} (g) u_{x}^{- 1} = σ_{c (x)} (g) (g \in G, μ -a.e. x) .$ In particular the measure class of the base and the measurable field of unitary equivalence classes of the fibre factor representations are invariants of $π$ . The literal parametrisation of these data by the quasi-dual $Q D (G)$ of $G$ (Bekka-de la Harpe Theorem 6.C.8) is not asserted here: it requires the Borel structure on the space of factor representations and the Borel quasi-dual map, which belong to the owner-held Glimm/smooth-dual branch of this pair.

Facts & Assumptions

Given: AC; the second-countable LCH group $G$ ; the separable strongly continuous unitary representation $(π, H)$ ; and two central decompositions of $π$ with data $(X, μ, π_{x}, U)$ and $(Y, ν, σ_{y}, V)$ .

[F1]

Transport of central decompositions: for two central decompositions of the same $π$ there are conull Borel sets $X_{0} \subseteq X$ , $Y_{0} \subseteq Y$ , a bimeasurable bijection $c : X_{0} \to Y_{0}$ with $c_{*} (μ ∣_{X_{0}})$ equivalent to $ν ∣_{Y_{0}}$ , and a measurable field $x \mapsto u_{x} : H_{x} \to K_{c (x)}$ of unitaries with $u_{x} π_{x} (g) u_{x}^{- 1} = σ_{c (x)} (g)$ for every $g \in G$ and $μ$ -almost every $x$ (Transport of central models and disintegration of intertwiners).

[F2]

A central decomposition of $π$ consists of a sigma-finite standard-Borel base $(X, B, μ)$ , a measurable Hilbert field with nonzero fibres almost everywhere, a measurable field $(π_{x})$ of strongly continuous unitary representations with $π_{x}$ factorial almost everywhere, and a unitary $U : H \to \int_{X}^{\oplus} H_{x} d μ (x)$ satisfying $U π (g) U^{- 1} = \int_{X}^{\oplus} π_{x} (g) d μ (x)$ , $U Z (π (G)^{''}) U^{- 1} = D$ and the corresponding commutant identities (Central decomposition into factor representations).

[F3]

A factor representation is one for which the centre of the generated von Neumann algebra is scalar; factoriality is preserved by unitary equivalence (Factor (primary) representations).

[F4]

AC is the stated hypothesis, inherited by [F1] and [F2] (The Axiom of Choice).

Proof

technique · apply the transport lemma to two central decompositions of the same representation and read off the invariance statement

Given: AC; the representation $(π, H)$ ; the two central decompositions $(X, μ, π_{x}, U)$ and $(Y, ν, σ_{y}, V)$ in the sense of [F2].

1.1F1F2F4

Both $(X, μ, π_{x}, U)$ and $(Y, ν, σ_{y}, V)$ are central decompositions of the same $π$ , so the hypotheses of the transport lemma [F1] are satisfied; we may apply it directly to obtain conull Borel sets $X_{0} \subseteq X$ and $Y_{0} \subseteq Y$ , a bimeasurable bijection $c : X_{0} \to Y_{0}$ and a measurable field of unitaries $x \mapsto u_{x} : H_{x} \to K_{c (x)}$ with $u_{x} π_{x} (g) u_{x}^{- 1} = σ_{c (x)} (g)$ for every $g \in G$ and almost every $x$ .

2.1F1F3step 1.1

The measure-class statement $c_{*} (μ ∣_{X_{0}}) \sim ν ∣_{Y_{0}}$ is part of [F1], so the two decompositions agree up to the bimeasurable base isomorphism $c$ and the null-set modification encoded in $X_{0}, Y_{0}$ ; the fibre unitary equivalence almost everywhere is step 1.1, and it preserves factoriality of the fibres by [F3].

2.2F1step 1.1algebra

Invariance: if $(X^{'}, μ^{'}, π_{x}^{'}, U^{'})$ is a third central decomposition of $π$ , applying step 1.1 to the pairs $(X, Y)$ and $(Y, X^{'})$ gives bimeasurable base isomorphisms whose composition is again bimeasurable and preserves measure classes, and the corresponding measurable fields of unitaries compose fibrewise; hence the relation "is related to by a bimeasurable base isomorphism and a measurable field of fibre unitaries" is an equivalence relation on central decompositions of $π$ , and the measure class of the base together with the measurable field of unitary equivalence classes of the fibre factor representations is an invariant of $π$ .

3.1F1F2step 2.2∎

The literal parametrisation by the quasi-dual $Q D (G)$ is outside the present claim. The transport result identifies the two standard-Borel bases and supplies measurable fibre unitaries without assigning quasi-dual labels.

Boundary cases

If $π$ is a factor representation, both central decompositions are trivial over one-point bases and the transport map is the identity of those points. If one base has measure zero, then $H = {0}$ , contrary to $H \neq {0}$ in the central-decomposition theorem, so this case does not arise; conull subsets $X_{0}, Y_{0}$ are chosen nonempty when the base is nonempty. If the two bases have different cardinalities of atoms, the bimeasurable bijection $c$ matches the atoms and preserves the measure class, which forces the corresponding atomic weights to be equivalent; no equality of measures is claimed, only equivalence of measure classes. The statement is an almost-everywhere statement with respect to $μ$ ; the exceptional null set may depend on the pair of decompositions but is chosen once. Choice content is that of [F4].

Source qualifications

Bekka–de la Harpe, Chapter 6 §6.C, Theorem 6.C.8 and Definition 6.C.9, printed pp. 197–198, describe uniqueness over the quasi-dual. The present base-identification claim is proved locally from the central transport lemma; literal quasi-dual parametrization is outside its scope. Blackadar, Part III §III.1.6.4, printed p. 254, outlines the central decomposition and the fibre commutant identities rather than supplying this spatialization and uniqueness proof.

Depends on

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Sources

Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019; author-hosted complete book draft) (standard reference, not scraped)
Bruce Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras (author-hosted complete text) (standard reference, not scraped)