Exercises on Geometric Module

4.6.1. *Proof*. Let $H_X$ and $H_Y$ be X module and $Y$ module respectively. The more usual definition of the support of a bounded operator from $H_X$ to $H_Y$ in the literature is as follows: the support of $T$ is the complement of all those points $(y,x)\in Y\times X$ for which there exist $f,g\in C_0(X)$ such that $g(y)f(x)\neq 0$ and $gTf=0$. Show that this definition is equivalent to Definition 4.1.7.

$\newcommand{\supp}{\operatorname{supp}}$
*Idea*. Take $X=Y$ and $T=\operatorname{id}$.

*Proof*. Let $U=\supp f:=\{x\in X| f(x)\neq 0\}$ and $V=\supp g$ respectively, then $U$ and $V$ are open subsets of $X$ and $Y$. Moreover,
\begin{align*}
& \chi_VT\chi_U=\frac{\chi_V}{g}gTf\frac{\chi_U}{f}\\
& gTf=g\chi_VT\chi_U f,
\end{align*}
where $\frac{\chi_U}{f}:=
\begin{cases}
\frac{1}{f(x)}, & x\in U\\
0, & \mbox{others}
\end{cases}.$

*Remark*. $\chi_{\supp f}$ is a projection in $B(H_X)$ whose range is the norm closure of $fH_X$.

Borel functional calculus on a locally compact Hausdorff space

**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]\[email protected]{–>}[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}