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(DE) ̸= DE, where DE is the canonical MASA in C* (E), then α(DE) ̸= wDEw * for all unitaries w ∈ C* (E). That is, the two MASAs DE and α(DE) of C* (E) are outer but not inner conjugate. 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.