On Conjugacy of MASAs in Graph $C^*$-Algebras

TL;DR

This paper investigates the conjugacy properties of maximal abelian subalgebras (MASAs) in graph $C^*$-algebras, showing that certain automorphisms produce MASAs that are outer but not inner conjugate, revealing structural rigidity.

Contribution

It establishes conditions under which MASAs in graph $C^*$-algebras are outer conjugate but not inner conjugate, extending results to non quasi-free automorphisms via graph isomorphisms.

Findings

Automorphisms fixing vertices can produce MASAs that are outer but not inner conjugate.

Changing the underlying graph can extend the applicability to non quasi-free automorphisms.

The result demonstrates a form of structural rigidity in the automorphism group of graph $C^*$-algebras.

Abstract

For a large class of finite graphs $E$, we show that whenever $\alpha$ is a vertex-fixing quasi-free automorphism of the corresponding graph $C^*$-algebra $C^*(E)$ such that $\alpha({\mathcal D}_E) \neq{\mathcal D}_E$, where ${\mathcal D}_E$ is the canonical MASA in $C^*(E)$, then $\alpha({\mathcal D}_E)\neq w{\mathcal D}_E w^*$ for all unitaries $w\in C^*(E)$. That is, the two MASAs ${\mathcal D}_E$ and $\alpha({\mathcal D}_E)$ of $C^*(E)$ are outer but not inner conjugate. Passing to an isomorphic $C^*$-algebra by changing the underlying graph makes this result applicable to certain non quasi-free automorphisms as well.