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Direct integrals transport along bimeasurable base isomorphisms

Statement

Assume AC. Let $(X, B_{X}, μ)$ and $(Y, B_{Y}, ν)$ be $σ$ -finite standard Borel measure spaces, let $c : X \to Y$ be a bimeasurable bijection with $ν = c_{*} μ$ , and let $(H_{y})_{y \in Y}$ be a measurable complex Hilbert field over $(Y, ν)$ with direct integral $\int_{Y}^{\oplus} H_{y} d ν (y)$ . Then $x \mapsto H_{c (x)}$ is a measurable Hilbert field over $(X, μ)$ with the pulled-back fundamental family, and pullback of sections $ξ \mapsto ξ \circ c$ is a unitary $c^{*} : \int_{Y}^{\oplus} H_{y} d ν (y) ⟶ \int_{X}^{\oplus} H_{c (x)} d μ (x)$ that intertwines multiplication by $f \in L^{\infty} (Y, ν)$ with multiplication by $f \circ c$ . A decomposable operator field $(T_{y})$ over $Y$ corresponds to the decomposable field $x \mapsto T_{c (x)}$ over $X$ with the same essential norm and the same fibrewise adjoint and product identities.

Facts & Assumptions

Given: AC, $σ$ -finite standard Borel measure spaces $(X, μ)$ , $(Y, ν)$ , a bimeasurable bijection $c : X \to Y$ with $ν = c_{*} μ$ , and a measurable Hilbert field $(H_{y}, e_{n} (y))_{y \in Y}$ with countable fundamental family and direct integral $H_{Y} = \int_{Y}^{\oplus} H_{y} d ν (y)$ .

[F1]

The field datum means exactly: a separable Hilbert space $H_{y}$ for each $y$ , vectors $e_{n} (y)$ spanning a dense subspace of $H_{y}$ , Borel Gram coefficients $y \mapsto ⟨ e_{n} (y), e_{m} (y) ⟩$ ; a section is measurable when all coefficients $y \mapsto ⟨ ξ (y), e_{n} (y) ⟩$ are Borel, and sections are identified when they agree off a Borel null set (Measurable Hilbert field from a countable fundamental family).

[F2]

$H_{Y}$ is the quotient of the square-integrable measurable sections by almost-everywhere agreement, with inner product $⟨ [ξ], [η] ⟩ = \int_{Y} ⟨ ξ (y), η (y) ⟩ d ν (y)$ , and it is a Hilbert space (Direct integral of a measurable Hilbert field, Direct integrals of measurable Hilbert fields are Hilbert spaces).

[F3]

For a section, coefficient measurability is equivalent to measurability of all pairings with measurable sections; such sections are closed under measurable scalar combinations and pointwise norm limits, and $y \mapsto ∥ ξ (y) ∥$ is measurable (Measurable sections have measurable pointwise inner products).

[F4]

Precomposition with the Borel maps $c$ and $c^{- 1}$ preserves Borel measurability (Composition with a Borel measurable outer map preserves measurability, Standard Borel spaces, Measurable spaces and measurable sets).

[F5]

An operator field $(T_{y})$ is weakly measurable when its fundamental matrix coefficients are Borel; it is essentially bounded when ${ess sup}_{y} ∥ T_{y} ∥ < \infty$ , and a bounded operator on $H_{Y}$ is decomposable when it acts by such a field, $S [ξ] = [T ξ]$ (Measurable and decomposable operator fields).

Proof

technique · direct

Given: AC, the base spaces and the field of the statement, with fundamental family $(e_{n})$ over $Y$ .

1.1F1F4

Define $e_{n}^{c} (x) : = e_{n} (c (x))$ in $H_{c (x)}$ . The Gram coefficients $x \mapsto ⟨ e_{n}^{c} (x), e_{m}^{c} (x) ⟩_{H_{c (x)}} = (⟨ e_{n}, e_{m} ⟩_{H_{∙}}) \circ c (x)$ are Borel by [F1] and [F4], and for each $x$ the span of ${e_{n}^{c} (x)}$ equals the span of ${e_{n} (c (x))}$ , which is dense in $H_{c (x)}$ ; hence $x \mapsto H_{c (x)}$ with this pulled-back family is a measurable Hilbert field with countable fundamental family over $(X, μ)$ .

2.1step 1.1F3F4

A section $ξ$ over $Y$ has Borel coefficients $⟨ ξ, e_{n} ⟩$ if and only if the section $ξ \circ c$ over $X$ has Borel coefficients $⟨ ξ \circ c, e_{n}^{c} ⟩ = (⟨ ξ, e_{n} ⟩) \circ c$ : one direction is [F4], and the converse applies [F4] to $c^{- 1}$ , which is bimeasurable; by [F3] the same equivalence holds for all pairings, and $∥ ξ \circ c ∥ = ∥ ξ ∥ \circ c$ is measurable whenever $ξ$ is.

3.1step 1.1step 2.1F1

Change of variables: for every nonnegative Borel function $h$ on $Y$ , $\int_{X} h (c (x)) d μ (x) = \int_{Y} h d ν$ , because $ν = c_{*} μ$ is the pushforward; consequently for a measurable section $ξ$ one has $\int_{X} ∥ ξ (c (x)) ∥_{H_{c (x)}}^{2} d μ (x) = \int_{Y} ∥ ξ (y) ∥_{H_{y}}^{2} d ν (y)$ , so $ξ \circ c$ is square-integrable exactly when $ξ$ is.

4.1step 2.1step 3.1F2

Define $c^{*} [ξ] : = [ξ \circ c]$ on the direct integral. It is well defined on classes: if $ξ = η$ outside a Borel $ν$ -null set $N$ , then $ξ \circ c = η \circ c$ outside $c^{- 1} (N)$ , and $μ (c^{- 1} (N)) = ν (N) = 0$ ; it is complex-linear because the fibre operations are pointwise and pullback is linear; and it preserves inner products, $⟨ c^{*} [ξ], c^{*} [η] ⟩ = \int_{X} ⟨ ξ (c (x)), η (c (x)) ⟩ d μ (x) = \int_{Y} ⟨ ξ (y), η (y) ⟩ d ν (y) = ⟨ [ξ], [η] ⟩$ , by [step 3.1]. It is surjective: for a measurable square-integrable section $η$ over $X$ , the section $ξ : = η \circ c^{- 1}$ is measurable over $Y$ by [step 2.1] applied to $c^{- 1}$ and has $ξ \circ c = η$ and the same integral by [step 3.1]. Hence $c^{*}$ is a complex-linear surjective isometry between the two direct integrals, that is, a unitary.

4.2F5step 1.1step 2.1step 3.1

A weakly measurable, essentially bounded operator field $(T_{y})$ over $Y$ pulls back to the operator field $x \mapsto T_{c (x)}$ on the fibres $H_{c (x)}$ : its fundamental matrix coefficients are $(⟨ T_{∙} e_{n}, e_{m} ⟩) \circ c$ , Borel by [F4], so the pulled field is weakly measurable, and ${x : ∥ T_{c (x)} ∥ > t} = c^{- 1} ({y : ∥ T_{y} ∥ > t})$ has $μ$ -measure $ν ({y : ∥ T_{y} ∥ > t})$ , so the two operator-norm functions have the same essential supremum; moreover, for a square-integrable section $ξ$ over $Y$ , the pulled section $T_{c (\cdot)} ξ (c (\cdot)) = (T ξ) \circ c$ is the pullback of the square-integrable section $T ξ$ , so the decomposable action is transported. Fibrewise adjoint and product identities are preserved because for each $x$ the fibre operator is $T_{c (x)}$ itself, $(T_{c (x)})^{*} = (T^{*})_{c (x)}$ and $S_{c (x)} T_{c (x)} = (S T)_{c (x)}$ .

5.1step 4.1step 4.2F2algebra∎

Finally $c^{*}$ intertwines multiplication: for $f \in L^{\infty} (Y, ν)$ and a square-integrable section $ξ$ , $c^{*} (M_{f} [ξ]) = [f ξ \circ c] = [(f \circ c) (ξ \circ c)] = M_{f \circ c} (c^{*} [ξ])$ pointwise. Together with [step 4.1] and [step 4.2] this proves that $x \mapsto H_{c (x)}$ is a measurable field, that $c^{*}$ is a unitary intertwining the two multiplication algebras, and that decomposable fields transport with the same essential norm and fibrewise algebraic identities.

Depends on

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Sources

J. Dixmier, Von Neumann Algebras, Chapter II §1 (direct integrals and measurable fields) (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)