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Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

Published online by Cambridge University Press:  13 June 2019

Nathan Brownlowe
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW2006, Australia ([email protected])
Marcelo Laca
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BCV8W 3P4, Canada ([email protected])
Dave Robertson
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan2308, Australia ([email protected])
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics,University of Wollongong, NSW2522, Australia ([email protected])

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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