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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • Received by editor(s): October 15, 2008
  • Received by editor(s) in revised form: December 8, 2009
  • Published electronically: February 4, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2775826