Isomorphisms between Leavitt algebras and their matrix rings

G. Abrams; P. N. ánh; E. Pardo

paper

Isomorphisms between Leavitt algebras and their matrix rings

arXiv:math/0612552

Abstract

Let $K$ be any field, let $L_{n}$ denote the Leavitt algebra of type $(1, n - 1)$ having coefficients in $K$ , and let $M_{d} (L_{n})$ denote the ring of $d \times d$ matrices over $L_{n}$ . In our main result, we show that $M_{d} (L_{n}) ≅ L_{n}$ if and only if $d$ and $n - 1$ are coprime. We use this isomorphism to answer a question posed in \cite{PS} regarding isomorphisms between various C*-algebras. Furthermore, our result demonstrates that data about the $K_{0}$ structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple $K$ -algebras.

Final version. To appear in J. reine Angew. Math