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Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  13 June 2019

Nathan Brownlowe ,
Marcelo Laca ,
Dave Robertson  and
Aidan Sims
Nathan Brownlowe
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW2006, Australia (nathan.brownlowe@sydney.edu.au)
Marcelo Laca
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BCV8W 3P4, Canada (laca@math.uvic.ca)
Dave Robertson
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan2308, Australia (dave84robertson@gmail.com)
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics,University of Wollongong, NSW2522, Australia (asims@uow.edu.au)
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Abstract

We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalized gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.

Keywords

Graph C*-algebraToeplitz algebraKMS state

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2632 - 2641
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Copyright
Copyright © Royal Society of Edinburgh 2019

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