Isomorphisms between Leavitt algebras and their matrix rings

Abstract

Let K be any field, let Ln denote the Leavitt algebra of type (1,n – 1) having coefficients in K, and let Md(Ln) denote the ring of d × d matrices over Ln. In our main result, we show that Md(Ln) ≅ Ln if and only if d and n – 1 are coprime. We use this isomorphism to answer a question posed in [W. Paschke and N. Salinas, Matrix algebras over , Michigan Math. J. 26 (1979), 3–12.] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstrates that data about the K0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple K-algebras.