Let $A$ and $B$ be nonzero unital $C^*$-algebras, and let $A \odot B$ denote their algebraic $*$-[tensor product](/page/Tensor%20Product). Let $H$ and $K$ be complex Hilbert spaces, and let $\pi:A\to B(H)$ and $\rho:B\to B(K)$ be faithful unital $*$-representations. Define the faithful spatial seminorm associated to $\pi$ and $\rho$ as the map

\begin{align*} \|\cdot\|_{\min,\pi,\rho}:A\odot B\to [0,\infty) \end{align*}

given, for $x\in A\odot B$, by

\begin{align*} \|x\|_{\min,\pi,\rho}:=\|(\pi\odot\rho)(x)\|_{B(H\otimes K)}. \end{align*}

Here $\pi\odot\rho:A\odot B\to B(H\otimes K)$ is the algebraic tensor product $*$-representation. Faithful unital representations exist by the [Universal Representation Theorem](/theorems/8566).

Then $\|x\|_{\min,\pi,\rho}$ is independent of the chosen faithful unital representations $\pi$ and $\rho$. Hence the formula

\begin{align*} \|\cdot\|_{\min}:A\odot B\to [0,\infty), \qquad \|x\|_{\min}:=\|x\|_{\min,\pi,\rho} \end{align*}

defines a norm on $A\odot B$. This norm is a $C^*$-norm on the algebraic $*$-algebra $A\odot B$, meaning that

\begin{align*} \|x^*x\|_{\min}=\|x\|_{\min}^2 \end{align*}

for every $x\in A\odot B$.