20 lines
835 B
TeX
20 lines
835 B
TeX
\section{The Borel Functional Calculus}
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\label{section:borel-functional-calculus}
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\begin{definition}[Borel Functional Calculus]
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\label{definition:borel-functional-calculus}
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Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $x \in A$ be normal, then there exists a unique continuous unital *-homomorphism
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\[
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C(\sigma_A(x); \complex)^{**} \to A[x] \quad f \mapsto f(x)
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\]
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such that:
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\begin{enumerate}
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\item $\one(x) = 1_A$, $\text{Id}(x) = x$, and $\overline{\text{Id}}(x) = x^*$.
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\item The mapping $f \mapsto f(x)$ is continuous from the weak* topology on $C(\sigma_A(x); \complex)^{**}$ to the strong operator topology on $B(H)$.
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\end{enumerate}
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\end{definition}
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\begin{proof}
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Since the \autoref{definition:continuous-functional-calculus}
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\end{proof}
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