Isomorphisms between Leavitt algebras and their matrix rings

TL;DR

This paper characterizes when matrix rings over Leavitt algebras are isomorphic to the original algebra, revealing a coprimality condition, and applies this to classify certain purely infinite simple algebras.

Contribution

It establishes a precise criterion for isomorphisms between Leavitt algebras and their matrix rings based on coprimality, and uses this to address questions in C*-algebra theory.

Findings

${ m M}_d(L_n) ot o L_n$ unless $d$ and $n-1$ are coprime

K_0$-theoretic data suffices to distinguish these algebras

Answer to a question in C*-algebra isomorphisms

Abstract

Let $K$ be any field, let $L_n$ denote the Leavitt algebra of type $(1,n-1)$ having coefficients in $K$, and let ${\rm M}_d(L_n)$ denote the ring of $d \times d$ matrices over $L_n$. In our main result, we show that ${\rm M}_d(L_n) \cong L_n$ 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.