Let $X$ be a nonempty locally compact [Hausdorff space](/page/Hausdorff%20Space). Let $C(X)$ denote the complex [algebra of continuous functions](/theorems/197) $f:X\to\mathbb C$. Let $C_0(X)$ be the complex $C^*$-algebra of functions in $C(X)$ vanishing at infinity, equipped with the supremum norm and pointwise operations, and let $C_b(X)$ be the unital commutative $C^*$-algebra of bounded functions in $C(X)$. Let $M(C_0(X))$ denote the multiplier algebra of $C_0(X)$ in the double-centralizer sense. Then the map

\begin{align*} \Theta:C_b(X)\to M(C_0(X)),\quad g\mapsto (L_g,R_g) \end{align*}

where $L_g:C_0(X)\to C_0(X)$ and $R_g:C_0(X)\to C_0(X)$ are given by $L_g(f)=gf$ and $R_g(f)=fg$, is a unital $*$-isomorphism of commutative $C^*$-algebras. In particular,

\begin{align*} M(C_0(X))\cong C_b(X). \end{align*}