On Conjugacy of Subalgebras in Graph $C^{\ast }$-Algebras. II

We apply a method inspired by Popa’s intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^{\ast }$-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 ${\mathcal{D}}_n$.