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Isomorphisms of algebras associated with directed graphs

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Abstract.

Given countable directed graphs G and G′, we show that the associated tensor algebras (G) and (G′) are isomorphic as Banach algebras if and only if the graphs G are G′ are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G′, the free semigroupoid algebras and are isomorphic as dual algebras if and only if the graphs G are G′ are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.

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Authors and Affiliations

  1. Department of Mathematics, East Carolina University, Greenville, NC, 27858, USA

    Elias Katsoulis

  2. Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1

    David W. Kribs

Authors
  1. Elias Katsoulis
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  2. David W. Kribs
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Correspondence to Elias Katsoulis.

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Mathematics Subject Classification (2000): 47L80, 47L55, 47L40

Acknowledgments. We would like to thank the referee for several constructive suggestions on the initial draft and for bringing to our attention the work in [8,9]. The first author was partially supported by a research grant from ECU and the second author by an NSERC research grant and start up funds from the University of Guelph. We thank David Pitts for enlightening conversations and Alex Kumjian for helpful comments on the literature.

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Katsoulis, E., Kribs, D. Isomorphisms of algebras associated with directed graphs. Math. Ann. 330, 709–728 (2004). https://doi.org/10.1007/s00208-004-0566-6

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  • DOI: https://doi.org/10.1007/s00208-004-0566-6

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