On Conjugacy of Subalgebras of Graph C*-Algebras
Abstract
The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. 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(E)≠E, where E is the canonical MASA in C*(E), then α(E)≠ wE w* for all unitaries w∈ C*(E). That is, the two MASAs E and α(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. For the Cuntz algebras On, we find a criterion which guarantees that a polynomial automorphism moves the canonical UHF subalgebra to a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of rescaling of trace on diagonal projections.