Borel functional calculus on a locally compact Hausdorff space

**Update**. We can generalize the Riesz representation theorem to the case of $C_0(X)$ first.

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**Lemma**. Suppose $X$ is a compact Hausdorff space, $H$ is a Hilbert space and $\pi:C(X)\to B(H)$ is a *-homomorphism. Then $\pi$ extends to a *-homomorphism on $B(X)$, the set of all bounded Borel functions on $X$. (The extension is unique in the sense …)

**Theorem**. Suppose $X$ is a locally compact Hausdorff space, $H$ is a Hilbert space and $\pi:C_0(X)\to B(H)$ is a *-homomorphism. Then $\pi$ extends to a *-homomorphism on $B(X)$, the set of all bounded Borel functions on $X$.

*Proof*. The one-point complication $X_\infty$ of $X$ is a compact Hausdorff space, and there is a isomomorphism between C*-algebras:
\begin{align*}
& C(X_\infty)\cong C_0(X)^+\\
& f\mapsto (f|_X+f(\infty), f(\infty)),
\end{align*}
and
\begin{align*}
& \pi^+: C_0(X)^+\to B(H)\\
& (f,\lambda)\mapsto \pi(f)+\lambda,
\end{align*}
is a *-homomophism extending $\pi$.

\begin{xy}\xymatrix{ & & & B(X)\ar[ld]\ar@{–>}[ddd]\\
& & B(X_\infty)\ar[d]\ar[ddr]\\
C_0(X)\ar[r]\ar[rrrd]\ar[rrruu] & C(X_\infty)\ar[r]\ar[rrd]\ar[ru] & C(X_\infty)^{**}\ar[rd] \\
&&& B(H)}\end{xy}

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