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Measurable fields of von Neumann algebras and their direct integrals

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $(X, B, μ)$ be a sigma-finite standard-Borel measure space (Standard Borel spaces, Finite, sigma-finite, and semifinite measures) and let $(H_{x}, e_{n} (x))_{x \in X}$ be a measurable complex Hilbert field with a countable fundamental family (Measurable Hilbert field from a countable fundamental family). Write $H = \int_{X}^{\oplus} H_{x} d μ (x)$ for its direct-integral Hilbert space (Direct integral of a measurable Hilbert field, Direct integrals of measurable Hilbert fields are Hilbert spaces). For every $x$ , let $M_{x} \subseteq B (H_{x})$ be a unital concrete von Neumann algebra (Von Neumann algebras and commutants). The field $x \mapsto M_{x}$ is measurable if there is a sequence $(T^{(k)})_{k \geq 1}$ of weakly measurable operator fields (Measurable and decomposable operator fields) such that

$M_{x} = W^{*} (T_{x}^{(1)}, T_{x}^{(2)}, \dots)$

for $μ$ -almost every $x$ , where $W^{*}$ denotes the weak-operator closure of the unital $*$ -algebra generated by the listed operators.

For a measurable field $x \mapsto M_{x}$ , define its direct integral to be the set

$\int_{X}^{\oplus} M_{x} d μ (x) : = {\int_{X}^{\oplus} T_{x} d μ (x) : (T_{x})_{x \in X} is weakly measurable and essentially bounded, and T_{x} \in M_{x} for μ -almost every x} \subseteq B (H),$

where each induced operator is supplied by Measurable essentially bounded operator fields act decomposably. Define the diagonal algebra by

$D : = {M_{f} : f \in L^{\infty} (X, μ), (M_{f} ξ) (x) = f (x) ξ (x)},$

using the complex $L^{\infty}$ convention of Complex Lp classes and Euclidean test-function conventions. Then

$D \subseteq \int_{X}^{\oplus} M_{x} d μ (x) .$

The fibres may be zero-dimensional: when $H_{x} = {0}$ , unitality means $M_{x} = {0}$ and $I_{H_{x}} = 0$ .

Facts & Assumptions

[A1]

AC is an explicit hypothesis and dependency. The von Neumann algebra setup, direct-integral Hilbert space, and decomposable operator action inherit the exact AC uses stated by their suppliers (The Axiom of Choice).

[F1]

Under AC the direct-integral space $H$ is a complete Hilbert space (Direct integrals of measurable Hilbert fields are Hilbert spaces).

[F2]

A concrete von Neumann algebra is a unital weak-operator-closed $*$ -subalgebra of $B (H_{x})$ ; the zero Hilbert space is allowed with sole unital algebra ${0}$ (Von Neumann algebras and commutants).

[F3]

An operator field is weakly measurable when its fundamental matrix coefficients are measurable; an operator field is essentially bounded when ${ess sup}_{x} ∥ T_{x} ∥ < \infty$ (Measurable and decomposable operator fields).

[F4]

Every weakly measurable essentially bounded field induces a well-defined bounded decomposable operator on $H$ , acting on classes by $[ξ] \mapsto [x \mapsto T_{x} ξ (x)]$ (Measurable essentially bounded operator fields act decomposably).

[F5]

Every $f \in L^{\infty} (X, μ)$ is an almost-everywhere class of measurable complex functions with finite essential bound (Complex Lp classes and Euclidean test-function conventions).

[F6]

The fundamental vectors $e_{n}$ are measurable sections, and $x \mapsto ⟨ e_{n} (x), e_{m} (x) ⟩$ is Borel measurable (Measurable Hilbert field from a countable fundamental family).

[F7]

Products of measurable complex scalar functions are measurable (Arithmetic and lattice operations preserve measurability whenever they are defined, Complex Lp classes and Euclidean test-function conventions).

[F8]

The direct integral identifies sections equal outside a measurable null set (Direct integral of a measurable Hilbert field).

[F9]

The direct-integral inner product is the integral of the fibre inner products, with the fibre pairing linear in the first variable (Direct integral of a measurable Hilbert field, Real and complex inner-product spaces and their induced length).

[F10]

$L^{\infty} (X, μ)$ is closed under sums, scalar multiples, products, and complex conjugation (Complex Lp classes and Euclidean test-function conventions).

[F11]

The base is a standard-Borel space with a sigma-finite measure, as assumed in the definition (Standard Borel spaces, Finite, sigma-finite, and semifinite measures).

Proof

technique · direct

Given: The AC-qualified measurable Hilbert field, its direct-integral Hilbert space, the field $x \mapsto M_{x}$ , and its countable weakly measurable generating family.

1.1A1F1F4F11

By [F1], $H$ is a Hilbert space. For any weakly measurable essentially bounded operator field $(T_{x})$ with $T_{x} \in M_{x}$ almost everywhere, [F4] supplies a bounded operator $\int_{X}^{\oplus} T_{x} d μ (x) \in B (H)$ . Thus the displayed direct integral is a well-defined subset of $B (H)$ ; this definition does not assert that the set is weak-operator closed.

1.2F2F3F5F6F7

Fix $f \in L^{\infty} (X, μ)$ and choose a measurable representative. By [F5], some finite $C$ and measurable null set $N$ satisfy $∣ f (x) ∣ \leq C$ for every $x \notin N$ . Define $T_{x} : = f (x) I_{H_{x}}$ . Its fundamental matrix coefficients are $⟨ T_{x} e_{n} (x), e_{m} (x) ⟩ = f (x) ⟨ e_{n} (x), e_{m} (x) ⟩$ , measurable by [F6, F7]; its operator norm is at most $∣ f (x) ∣$ , including when $H_{x} = {0}$ , so it is essentially bounded. Since every $M_{x}$ is unital, $f (x) I_{H_{x}} \in M_{x}$ for every $x$ ; at a zero fibre this is $0 \in {0}$ . Hence this field satisfies the direct-integral membership conditions.

2.1F4F8F9F10step 1.2∎

By [F4], the field of step 1.2 induces a decomposable operator acting on square-integrable classes by $[ξ] \mapsto [x \mapsto f (x) ξ (x)]$ , which is exactly $M_{f}$ . If $f$ is changed on a measurable null set, [F4] gives the same induced operator, so $M_{f}$ depends only on its $L^{\infty}$ class. Pointwise action gives $M_{f} + M_{g} = M_{f + g}$ , $c M_{f} = M_{c f}$ , and $M_{f} M_{g} = M_{f g}$ ; the first-variable-linear integral pairing gives $⟨ M_{f} [ξ], [η] ⟩ = ⟨ [ξ], M_{\overline{f}} [η] ⟩$ , so $M_{f}^{*} = M_{\overline{f}}$ . Since $1 \in L^{\infty}$ and $M_{1} = I_{H}$ (including $I_{H} = 0$ when $H = {0}$ ), [F8] makes $D$ a unital $*$ -algebra. Each $M_{f}$ belongs to the displayed direct-integral set by step 1.2, proving $D \subseteq \int_{X}^{\oplus} M_{x} d μ (x)$ .

Remarks

The definition specifies a set of decomposable operators. Weak-operator closure of this set is a separate theorem for measurable fields; no closure assertion is built into the definition. The diagonal inclusion is proved locally above.

Sources

Bekka–de la Harpe, Unitary Representations of Groups, Duals, and Characters, Chapter 1 §1.I, Definition 1.I.1 (measurable fields), printed p. 69; Definition 1.I.4 and Example 1.I.5 (direct integrals and the diagonal algebra), printed pp. 69–70. Proposition 1.I.3 separately asserts weak-operator closure of the direct-integral set and refers its proof to Dixmier–von Neumann; this item does not use that result.

Depends on

Used by

Dependency tree · two levels

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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) (standard reference, not scraped)