Tensor products of Leavitt path algebras

Abstract

We compute the Hochschild homology of Leavitt path algebras over a field k. As an application, we show that L2 and L2 ⊗ L2 have different Hochschild homologies, and so they are not Morita equivalent; in particular, they are not isomorphic. Similarly, L∞ and L∞ ⊗ L∞ 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∗(L2) = K∗(L2 ⊗ L2) = 0 and K∗(L∞) = K∗(L∞ ⊗ L∞) = K∗(k).