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Quantum Symmetries of Graph C*-algebras

Published online by Cambridge University Press:  20 November 2018

Simon Schmidt
Affiliation:
Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany, e-mail : simon.schmidt@math.uni-sb.de, weber@math.uni-sb.de
Moritz Weber
Affiliation:
Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany, e-mail : simon.schmidt@math.uni-sb.de, weber@math.uni-sb.de

Abstract

The study of graph ${{C}^{*}}$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph ${{C}^{*}}$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph ${{C}^{*}}$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

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References

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