Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2011-05264-5
Published electronically: February 4, 2011
MathSciNet review: 2775826
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 16D70, 46L55

Retrieve articles in all journals with MSC (2010): 16D70, 46L55


Additional Information

Gene Abrams
Affiliation: Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933
MR Author ID: 190273
Email: abrams@math.uccs.edu

Mark Tomforde
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
MR Author ID: 687274
Email: tomforde@math.uh.edu

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.