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All finite transitive graphs admit a self-adjoint free semigroupoid algebra

Part of: Linear spaces and algebras of operators Graph theory

Published online by Cambridge University Press:  30 March 2020

Adam Dor-On  and
Christopher Linden
Adam Dor-On
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark (adoron@math.ku.dk)
Christopher Linden
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA (clinden2@illinois.edu)
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Abstract

In this paper we show that every non-cycle finite transitive directed graph has a Cuntz–Krieger family whose WOT-closed algebra is $B(\mathcal {H})$. This is accomplished through a new construction that reduces this problem to in-degree 2-regular graphs, which is then treated by applying the periodic Road Colouring Theorem of Béal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.

Keywords

Graph algebraCuntz KriegerFree semigroupoid algebraRoad colouringPeriodicCyclic decompositionDirected graphs

MSC classification

Primary: 47L55: Representations of (nonselfadjoint) operator algebras
Secondary: 47L15: Operator algebras with symbol structure 05C20: Directed graphs (digraphs), tournaments
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 391 - 406
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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