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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Isomorphism and Morita equivalence of graph algebras


Authors: Gene Abrams and Mark Tomforde
Journal: Trans. Amer. Math. Soc. 363 (2011), 3733-3767
MSC (2010): Primary 16D70, 46L55
Published electronically: February 4, 2011
MathSciNet review: 2775826
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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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Additional Information

Gene Abrams
Affiliation: Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933
Email: abrams@math.uccs.edu

Mark Tomforde
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: tomforde@math.uh.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05264-5
PII: S 0002-9947(2011)05264-5
Keywords: Graph, Leavitt path algebra, graph $C^{*}$-algebra, Morita equivalence
Received by editor(s): October 15, 2008
Received by editor(s) in revised form: December 8, 2009
Published electronically: February 4, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.