Let $X$ and $Y$ be compact Hausdorff spaces. Let $C(X)$, $C(Y)$, and $C(X\times Y)$ denote the complex $C^*$-algebras of continuous complex-valued functions on $X$, $Y$, and $X\times Y$, respectively, with pointwise operations, complex conjugation as involution, and the supremum norm. Let $C(X)\odot C(Y)$ denote the algebraic $*$-[tensor product](/page/Tensor%20Product) and let $C(X)\otimes_{\min} C(Y)$ denote its completion for the minimal $C^*$-tensor norm. There is a unique [linear map](/page/Linear%20Map)

\begin{align*} \Phi_0:C(X)\odot C(Y)\to C(X\times Y) \end{align*}

such that, for every $f\in C(X)$, every $g\in C(Y)$, and every $(x,y)\in X\times Y$,

\begin{align*} \Phi_0(f\otimes g)(x,y)=f(x)g(y). \end{align*}

This map is a unital $*$-homomorphism and extends uniquely to an isometric $*$-isomorphism

\begin{align*} \Phi:C(X)\otimes_{\min}C(Y)\to C(X\times Y). \end{align*}