On Conjugacy of Subalgebras in Graph $C^*$-Algebras. II

TL;DR

This paper investigates the inner conjugacy of maximal abelian subalgebras (MASAs) in graph $C^*$-algebras, providing new proofs and identifying classes of MASAs not conjugate to the diagonal, using Popa's intertwining technique.

Contribution

It introduces a novel application of Popa's intertwining-by-bimodules method to graph $C^*$-algebras, proving non-conjugacy of certain MASAs and expanding understanding of their structure.

Findings

Diagonal MASA not inner conjugate to its automorphic images

Identification of MASAs in Cuntz algebras not conjugate to the diagonal

Extension of results to non quasi-free automorphisms

Abstract

We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal D}_n$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal O}_n$ that are not inner conjugate to the diagonal ${\mathcal D}_n$.