On Conjugacy of MASAs in Graph $C^{\ast }$-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 $C^{\ast }$-algebra $C^{\ast }(E)$ such that $\alpha ({\mathcal{D}}_E)\ne {\mathcal{D}}_E$, where ${\mathcal{D}}_E$ is the canonical MASA in $C^{\ast }(E)$, then $\alpha ({\mathcal{D}}_E)\ne w{\mathcal{D}}_E{w}^{\ast }$ for all unitaries $w\in C^{\ast }(E)$. That is, the two MASAs ${\mathcal{D}}_E$ and $\alpha ({\mathcal{D}}_E)$ of $C^{\ast }(E)$ are outer but not inner conjugate. Passing to an isomorphic $C^{\ast }$-algebra by changing the underlying graph makes this result applicable to certain non quasi-free automorphisms as well.

9 pages

References in corpus (1)

• Cartan subalgebras and the UCT problem, II