Isomorphisms between Leavitt algebras and their matrix rings

Details Isomorphisms between Leavitt algebras and their matrix rings Isomorphisms between Leavitt algebras and their matrix rings

2006-12-19

arxiv.org/abs/math/0612552v2

Mathematics

Rings and Algebras

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

Scientific paper

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.

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