Definition 39.5.3 (Integration Against Spectral Measure).label Let $X$ be a compact Hausdorff space, $H$ be a complex Hilbert space, $E: \cb_{X} \to B(H)$ be a spectral measure relative to $H$, $\mathscr{E}\subset M_{R}(X; \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$, and
Then, $\mathscr{E}^{*}$ admits a unique weak*-continuous involution and a unique separately weak*-continuous product, such that $\mathscr{E}^{*}$ is a commutative unital $C^{*}$-algebra, and $J$ is a unital *-homomorphism.
For each $\phi \in \mathscr{E}^{*}$, let $I_{E}(\phi) \in B(H)$ be the operator defined by
then
- (1)
$I_{E}$ is a contraction from $\mathscr{E}^{*}$ to $B(H)$.
- (2)
$I_{E}$ is continuous from the weak*-topology on $\mathscr{E}^{*}$ to the weak operator topology on $B(H)$.
- (3)
$I_{E}$ is an injective unital *-homomorphism.
For any $\phi \in \mathscr{E}^{*}$, $I_{E}(\phi) = \int_{X} \phi dE$ is the integral of $\phi$ with respect to $E$.
Proof. (1): Let $\phi \in \mathscr{E}^{*}$ and $x, y \in H$, then by Lemma 39.5.2,
Since the above holds for all $x, y \in H$, $I_{E}(\phi) \in B(H)$ with $\norm{I_E(\phi)}_{B(H)}\le \norm{\phi}_{\mathscr{E}^{*}}$.
(2): For each $x, y \in H$, $E_{x, y}\in \mathscr{E}$. Since $\angles{\int \phi dE \cdot x, y}_{H}= \dpn{E_{x, y}, \phi}{\mathscr{E}}$ for every $\phi \in \mathscr{E}^{*}$, $I_{E}$ is continuous from the weak* topology on $\mathscr{E}^{*}$ to the weak operator topology on $B(H)$.
(3): By Lemma 26.6.4, the simple functions $\Sigma(X; \complex)$ are uniformly dense in the bounded Borel functions $B^{\infty}(X; \complex)$. Since
- (i)
$I_{E}$ restricted to $J(\Sigma(X; \complex))$ is a *-homomorphism.
- (ii)
Multiplication and conjugation are continuous in the uniform norm on $B^{\infty}(X; \complex)$
- (iii)
Composition and adjunction are continuous in the operator norm on $B(H)$
the map $I_{E}$ restricted to $J(B^{\infty}(X; \complex))$ is a *-homomorphism by continuity.
By Goldstine’s Theorem, $C(X; \complex) \subset B^{\infty}(X; \complex)$ is weak*-dense in $C(X; \complex)^{**}$, so $J(C(X; \complex))$ is weak*-dense in $\mathscr{E}^{*}$. As
- (i)
$I_{E}$ restricted to $J(B^{\infty}(X; \complex))$ is a *-homomorphism.
- (ii)
The involution $\phi \mapsto \ol \phi$ is weak*-continuous on $\mathscr{E}^{*}$.
- (iii)
The adjunction $T \mapsto T^{*}$ is weak-operator continuous on $B(H)$.
- (iv)
The product $(\phi, \psi) \mapsto \phi \psi$ is separately weak*-continuous on $\mathscr{E}^{*}$.
- (v)
The composition $(S, T) \mapsto ST$ is separately weak-operator continuous on $B(H)$.
the map $I_{E}$ is a *-homomorphism by the weak* to weak-operator continuity established in (2). Since $E(X) = I_{B(H)}$, $I_{E}$ is a unital *-homomorphism.
Finally, let $\phi \in \mathscr{E}^{*}$ with $I_{E}(\phi) = 0$, then $\dpn{I_E(\phi)x, y}{H}= \dpn{E_{x, y}, \phi}{\mathscr{E}}= 0$ for all $x, y \in H$. As $\mathscr{E}$ is the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$, $\phi = 0$. Therefore $I_{E}$ is an injective unital *-homomorphism.$\square$
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