TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Pvm integral is a star homomorphism

Statement

Assume Countable Choice. Let $(X, Σ)$ be a measurable space, let $H$ be a nonzero complex Hilbert space, let $E$ be a projection valued measure on $(X, Σ)$ , and for a bounded measurable $f$ write $Φ_{E} (f) = \int f d E$ for the operator of Bounded borel pvm integral. Then:

$Φ_{E}$ is linear and unital: $Φ_{E} (1_{X}) = I$ and $Φ_{E} (a f + b g) = a Φ_{E} (f) + b Φ_{E} (g)$ for bounded measurable $f, g$ and $a, b \in C$ ;
$Φ_{E}$ is multiplicative: $Φ_{E} (f g) = Φ_{E} (f) Φ_{E} (g)$ ;
$Φ_{E}$ preserves conjugation: $Φ_{E} (\overline{f}) = Φ_{E} (f)^{*}$ ;
if $(f_{n})$ are bounded measurable with $\sup_{n} ∥ f_{n} ∥_{\infty} < \infty$ , $f$ is bounded measurable, and $f_{n} \to f$ pointwise $E$ -almost everywhere, meaning $E_{x}$ -almost everywhere for every $x \in H$ , then $Φ_{E} (f_{n}) \to Φ_{E} (f)$ in the strong operator topology.

Facts & Assumptions

[A1]

$Φ_{E} (g)$ is the unique operator with $⟨ Φ_{E} (g) x, y ⟩ = \int g d E_{x, y}$ for all $x, y$ , it satisfies $∥ Φ_{E} (g) ∥ \leq ∥ g ∥_{\infty}$ and $∥ Φ_{E} (g) x ∥^{2} = \int ∣ g ∣^{2} d E_{x}$ , and it is the norm limit of $\int s_{n} d E$ for any complex simple $s_{n} \to g$ uniformly (Bounded borel pvm integral).

[A2]

The simple integral of $s = \sum_{j} c_{j} 1_{D_{j}}$ over a disjoint measurable cover is $\sum_{j} c_{j} E (D_{j})$ (Integral of a simple function against a pvm), independently of the presentation; it has the scalar pairing and quadratic identities (Simple pvm integral is representation independent). Complex simple functions have finite measurable range (Complex simple functions as finite sums of measurable indicators).

[A3]

$E_{y, x} = \overline{E_{x, y}}$ , $E_{x}$ is a positive measure of mass $∥ x ∥^{2}$ , and $E_{x} (B) = ⟨ E (B) x, x ⟩$ (Scalar and complex measures from a pvm).

[A4]

Each $E (B)$ is self-adjoint and idempotent, $E (\emptyset) = 0$ , $E (X) = I$ and $E (B \cap C) = E (B) E (C)$ (Projection valued measure).

[A5]

Dominated convergence: if $g_{n} \to g$ pointwise almost everywhere and $∣ g_{n} ∣ \leq C$ for an integrable constant $C$ , then $\int g_{n} d μ \to \int g d μ$ (Dominated convergence).

[A6]

The adjoint is conjugate-linear on operator sums and norm-preserving, $∥ S^{*} ∥ = ∥ S ∥$ , and operator multiplication is norm-continuous, $∥ S T ∥ \leq ∥ S ∥ ∥ T ∥$ (Hilbert-adjoint identities, Composition satisfies |ST|\le|S|,|T|).

[A7]

Countable Choice is the declared standing hypothesis of this block of the page (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: A measurable space $(X, Σ)$ , a nonzero complex Hilbert space $H$ , a projection valued measure $E$ , bounded measurable functions $f, g$ with uniformly approximating complex simple functions $s_{n} \to f$ , $t_{n} \to g$ , and scalars $a, b$ .

1.1A1A2A4

Unitality: $1_{X}$ is simple, and $Φ_{E} (1_{X}) = E (X) = I$ by the definition of the simple integral and $E (X) = I$ .

1.2A2

Linearity on simple functions: presenting $s$ and $t$ over a common refinement of their disjoint normal forms, $\int (a s + b t) d E = a \int s d E + b \int t d E$ because both sides are the corresponding coefficient-weighted sum of the same projection values.

1.3A2A4

Multiplicativity on simple functions: over a common disjoint normal form $s = \sum_{j} c_{j} 1_{D_{j}}$ , $t = \sum_{j} d_{j} 1_{D_{j}}$ one has $s t = \sum_{j} c_{j} d_{j} 1_{D_{j}}$ and $(\int s d E) (\int t d E) = \sum_{j, k} c_{j} d_{k} E (D_{j}) E (D_{k}) = \sum_{j} c_{j} d_{j} E (D_{j}) = \int s t d E$ , because $E (D_{j}) E (D_{k}) = E (D_{j} \cap D_{k})$ vanishes for $j \neq k$ and equals $E (D_{j})$ for $j = k$ .

1.4A1A2A4A6

Conjugation on simple functions: self-adjointness of the projection values and conjugate-linearity of the adjoint give $(\int s d E)^{*} = (\sum_{j} c_{j} E (D_{j}))^{*} = \sum_{j} \overline{c_{j}} E (D_{j}) = \int \overline{s} d E$ . The bounded integral agrees with the simple integral by taking a constant approximating sequence.

2.1A1A6step 1.2step 1.3step 1.4

Linearity, multiplicativity and conjugation pass to uniform limits: if $s_{n} \to f$ and $t_{n} \to g$ uniformly then $a s_{n} + b t_{n} \to a f + b g$ , $s_{n} t_{n} \to f g$ and $\overline{s_{n}} \to \overline{f}$ uniformly, and A1 gives convergence of the simple integrals to the integrals of each of these limits, so $Φ_{E} (a f + b g) = a Φ_{E} (f) + b Φ_{E} (g)$ , $Φ_{E} (f g) = Φ_{E} (f) Φ_{E} (g)$ and $Φ_{E} (\overline{f}) = Φ_{E} (f)^{*}$ by taking norm limits and using norm continuity of the adjoint.

3.1step 2.1A1A3A5

Strong convergence: if $\sup_{n} ∥ f_{n} ∥_{\infty} \leq C < \infty$ and $f_{n} \to f$ $E$ -almost everywhere, then for each $x$ the functions $∣ f_{n} - f ∣^{2}$ converge to $0$ $E_{x}$ -almost everywhere and are dominated by the constant $(C + ∥ f ∥_{\infty})^{2}$ , which is integrable for the finite measure $E_{x}$ ; hence $∥ Φ_{E} (f_{n}) x - Φ_{E} (f) x ∥^{2} = \int ∣ f_{n} - f ∣^{2} d E_{x} \to 0$ .

4.1step 1.1step 2.1step 3.1A7∎

$Φ_{E}$ is a unital star homomorphism on the bounded measurable functions, and bounded pointwise $E$ -almost everywhere convergence with a uniform bound implies strong convergence of the operators.

Pvm integral is a star homomorphism

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources