Let $A$ and $B$ be complex $C^*$-algebras, let $H$ and $K$ be complex Hilbert spaces, and let $\pi:A\to \mathcal{L}(H)$ and $\rho:B\to \mathcal{L}(K)$ be $*$-representations. Let $A\odot B$ denote the algebraic [tensor product](/page/Tensor%20Product) $*$-algebra, let $H\odot K$ denote the algebraic tensor product [inner product space](/page/Inner%20Product%20Space), and let $H\otimes K$ denote its [Hilbert space](/page/Hilbert%20Space) completion. Then there is a unique [linear map](/page/Linear%20Map) $\pi\odot\rho:A\odot B\to \mathcal{L}(H\otimes K)$ satisfying

\begin{align*} (\pi\odot\rho)(a\otimes b)=\pi(a)\otimes \rho(b) \end{align*}

for every $a\in A$ and $b\in B$, where $\pi(a)\otimes\rho(b)$ is the bounded operator on $H\otimes K$ induced by its action on elementary tensors in $H\odot K$. Moreover, this map is a $*$-homomorphism.