Definition 6.1.3.label Let $\OM$ be a locally compact space. The set $C_{\infty}(\OM)$ of all continuous functions on $\OM$ vanishing at infinity is a $C^{*}$-algebra the following structure
\begin{gather*}(\lam x+\mu y)(\om)=\lam x(\om)+\mu \\ (xy)(\om)=x(\om)y(\om)\\ x^{*}(\om)=\cl{x(\om)}\\ \norm{x}{}=\sup\curl{\abs{x(\om)}:\om\in \OM}\end{gather*}
for every $x,y\in C_{\infty}(\OM)$, $\lam,\mu\in \com$ and $\om\in \OM$. The $C^{*}$-algebra $C_{\infty}(\OM)$ is abelian. The algebra $C_{\infty}(\OM)$ has an identity if and only if $\OM$ is compact and we shall denote $C_{\infty}(\OM)=C(\OM)$.
If $\hi$ is a Hilbert space, then the Banach algebra $\LIN{\hi}$ of all bounded operators on $\hi$ is a $C^{*}$-algebra with the involution $x\mto x^{*}$ defined as the adjoint operator $x^{*}$ of $x$. If the dimension of $\hi$ is greater than one, then $\LINH$ is not abelian.