Tensor products of Leavitt path algebras

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_{\mathrm{\infty }}$ and $L_{\mathrm{\infty }}\otimes L_{\mathrm{\infty }}$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_{\ast }(L_2)=K_{\ast }(L_2\otimes L_2)=0$ and $K_{\ast }(L_{\mathrm{\infty }})=K_{\ast }(L_{\mathrm{\infty }}\otimes L_{\mathrm{\infty }})=K_{\ast }(k)$.