On conjugacy of subalgebras in graph C*-algebras. II

We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph -algebras. First, we give a new proof of non-inner conjugacy of the diagonal MASA to its non-trivial image under a quasi-free automorphism, where 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 that are not inner conjugate to the diagonal.