Let $A$, $B$ be $C^\ast$-algebras. We say a representation of $A$: $\pi: A\rightarrow B(H)$ is irreducible if the only closed vector subspaces of $H$ that are invariant for $A$ are $0$ and $H$.
Now, Let $\pi: A\rightarrow B(H)$ and $\sigma: B\rightarrow B(K)$ be irreducible representations of $A$ and $B$, respectively.
My question is whether the tensor product $\pi\otimes \sigma: A\odot B\rightarrow B(H\otimes K)$ induced by $\pi(a)\otimes\sigma(b)(x\otimes y)=\pi(a)(x)\otimes\sigma(b)(y)$ is irreducible as well? Here $A\odot B$ is algebraic tensor product and irreduciblity also makes sense for $\ast$-algebras.
I guess it is true but I cannot prove it.
Thank you for all helps!