Isomorphism and Morita equivalence of graph algebras

Abrams, Gene; Tomforde, Mark

doi:10.1090/S0002-9947-2011-05264-5

Isomorphism and Morita equivalence of graph algebras
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by Gene Abrams and Mark Tomforde PDF

Trans. Amer. Math. Soc. 363 (2011), 3733-3767 Request permission

Abstract:

For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\mathbb {C}}(E)$ and the graph $C^*$-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring $*$-homomorphisms, algebra homomorphisms, and algebra $*$-homomorphisms between $L_{\mathbb {C}}(E)$ and $L_{\mathbb {C}}(F)$. We prove that in certain situations isomorphisms between $L_{\mathbb {C}}(E)$ and $L_{\mathbb {C}}(F)$ yield $*$-isomorphisms between the corresponding $C^*$-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that $*$-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\mathbb {C}}(E)$ and $L_{\mathbb {C}}(F)$ in specific cases. The relationship between Leavitt path algebras and graph $C^*$-algebras is also explored in the context of Morita equivalence.

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