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

For a large class of finite graphs $E$, we show that whenever $α$ is a vertex-fixing quasi-free automorphism of the corresponding graph $C^*$-algebra $C^*(E)$ such that $α({\mathcal D}_E) \neq{\mathcal D}_E$, where ${\mathcal D}_E$ is the canonical MASA in $C^*(E)$, then $α({\mathcal D}_E)\neq w{\mathcal D}_E w^*$ for all unitaries $w\in C^*(E)$. That is, the two MASAs ${\mathcal D}_E$ and $α({\mathcal D}_E)$ 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.

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