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Central disintegration: fibre commutant, centre and factoriality

Statement

Assume the Axiom of Choice. Let $G$ be a second-countable locally compact group, $(π, H)$ a separable strongly continuous unitary representation, and let $U : H \to \int_{X}^{\oplus} H_{x} d μ (x)$ be a direct-integral model with diagonal algebra $D$ such that $U Z (π (G)^{''}) U^{- 1} = D$ (that is, $U$ diagonalises the centre). Put $M : = U π (G)^{''} U^{- 1}$ . Then $D \subseteq M \subseteq D^{'}$ , and, for the measurable field of von Neumann algebras $x \mapsto M_{x} : = π_{x} (G)^{''}$ generated by the disintegration $π = \int^{\oplus} π_{x} d μ$ of Disintegration of a separable group representation over a commuting diagonal algebra, one has $M = \int_{X}^{\oplus} M_{x} d μ (x), M^{'} = \int_{X}^{\oplus} M_{x}^{'} d μ (x), Z (M) = \int_{X}^{\oplus} Z (M_{x}) d μ (x) .$ Consequently $Z (M_{x}) = C I_{H_{x}}$ for almost every $x$ , thus $π_{x}$ is a factor representation for almost every $x$ in the Borel stratum $X_{+} : = {x : H_{x} \neq {0}}$ ; and if two measurable fields of unital von Neumann algebras have the same direct integral they agree almost everywhere, so the centre field is intrinsically determined.

Facts & Assumptions

[F1]

Disintegration over the diagonal algebra gives a measurable field $π_{x}$ , and countably many bounded integrated operators $A_{j}$ generating $M$ whose fibres generate $M_{x} = π_{x} (G)^{''}$ (Disintegration of a separable group representation over a commuting diagonal algebra).

[F2]

A measurable von Neumann algebra field has measurable commutant and centre fields, a von Neumann direct integral with fibrewise commutant and centre, and equal direct integrals imply equality of fields almost everywhere (Measurable fields of von Neumann algebras have measurable commutants and centers, Measurable fields of von Neumann algebras and their direct integrals).

[F3]

A concrete von Neumann algebra equals its double commutant (The double commutant theorem for concrete von Neumann algebras). The spectral model realizes the centre as the scalar diagonal algebra (Spectral multiplicity model for separably acting abelian von Neumann algebras). AC is The Axiom of Choice.

[F4]

A factor representation has a nonzero Hilbert carrier and scalar centre of its generated algebra (Factor (primary) representations). The zero-fibre stratum is Borel: it is the intersection of the Borel zero sets of the fundamental norms, whose vectors have dense fibrewise span.

Proof

technique · direct

Given: The hypotheses and notation of the Statement, including AC.

1.1F1F2F3givenalgebra

Since $D = Z (M)$ , $D \subseteq M \subseteq D^{'}$ . Put $A = \int_{X}^{\oplus} M_{x} d μ$ . By [F2] this is a von Neumann algebra. Each integrated generator $A_{j}$ belongs to $A$ by [F1], so $M \subseteq A$ . Choose a countable dense set $(g_{n})$ in $G$ . Every $B \in M^{'}$ commutes with $D$ , hence is decomposable; its fibres commute with $π_{x} (g_{n})$ almost everywhere for each $n$ by uniqueness of decomposable representatives. Off their countable union of null sets, they commute with all $π_{x} (g)$ by strong continuity and boundedness of $B_{x}$ . Thus $B_{x} \in M_{x}^{'}$ almost everywhere. If $T \in A$ , its fibres commute with those of each such $B$ ; consequently $T \in (M^{'})^{'} = M$ . The exceptional set may depend on $B$ , which is harmless: membership in $M^{''}$ requires commutation with each global $B$ , not a common fibre representative for all $B$ . Hence $M = A$ .

2.1F2F4step 1.1algebra∎

Apply [F2] to this equality to obtain $M^{'} = \int_{X}^{\oplus} M_{x}^{'} d μ$ and $Z (M) = \int_{X}^{\oplus} Z (M_{x}) d μ$ . The scalar field $C I_{H_{x}}$ is measurable and its integral is exactly $D = Z (M)$ . The equality-of-integrals clause of [F2] therefore gives $Z (M_{x}) = C I_{H_{x}}$ almost everywhere. On the Borel nonzero-fibre stratum $X_{+}$ this makes $π_{x}$ factorial by [F4]; on zero fibres both algebras are ${0}$ , and no nonzero factor representation is asserted. The same clause gives the final intrinsic-field assertion for any two measurable fields.

Boundary and source qualifications

AC is inherited from disintegration, spectral and measurable-field suppliers and supplies a countable dense enumeration of G. The zero-fibre stratum is Borel because all fundamental vectors vanish there. It may have positive measure and contributes the zero algebra; factoriality is asserted only on the nonzero-fibre stratum. If the total space is zero, sigma-finiteness and the fundamental family force the nonzero-fibre stratum to be null, so only its factoriality assertion is vacuous; the algebra identities still hold. No everywhere selector or uncountable union of exceptional null sets is used. No source citation replaces a local supplier proof. The referenced complete Bekka–de la Harpe PDF, pp. 195–202, and Blackadar PDF pp. 255–262 were consulted for the central/type-I architecture; Blackadar explicitly outlines the direct-integral theory and refers technical details elsewhere. The measurable and spatial steps here use the proved local suppliers named above.

Depends on

Used by

Central decomposition into factor representations Theorem

Dependency tree · two levels

90 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)