Definition 37.4.4 (Borel Functional Calculus).label Let $H$ be a complex Hilbert space, $T \in B(H)$ be normal, and $A \subset B(H)$ be the smallest von Neumann algebra acting on $H$ containing $T$ and $I$, then there exists a unique unital *-homomorphism
such that:
- (1)
$\one(T) = I$, $\text{Id}(T) = T$, $\ol{\text{Id}}(T) = T^{*}$.
- (2)
The mapping $\phi \mapsto \phi(T)$ is continuous from the weak* topology on $C(\sigma_{B(H)}(T); \complex)^{**}$ to the weak operator topology on $A$.
Moreover, there exists a unique spectral measure $E: \cb_{\sigma_{B(H)}(T)}\to A$ such that $\phi(T) = \int_{\sigma_{B(H)}(T)}\phi dE$ for all $\phi \in C(\sigma_{B(H)}(T); \complex)^{**}$.
Proof. By the Spectral Theorem applied to $B(H)[T]$, there exists a unique spectral measure $E$ on $\sigma_{B(H)}(T)$ such that the mapping
extends the inverse Gelfand transform $\Gamma_{B(H)[T]}^{-1}: C(\sigma_{B(H)}(T); \complex) \to B(H)[T]$.
For each $\phi \in C(\sigma_{B(H)}(T); \complex)^{**}$, let $\phi(T) = \int_{\sigma_{B(H)}(T)}\phi dE$, then the mapping $\phi \mapsto \phi(T)$ is continuous from the weak* topology on $C(\sigma_{B(H)}(T); \complex)^{**}$ to the weak operator topology on $A$ by Definition 37.4.2.
Finally, by uniqueness of the continuous functional calculus, Goldstine’s Theorem, and (2), the mapping $\phi \mapsto \phi(T)$ is unique.$\square$
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