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(D_E) ≠ D_E where D_E is the canonical MASA in C^*(E), then α(D_E) ≠ ωD_Eω^* for all unitaries ω ∈ C^* (E). That is, the two MASA_s D_E and α(D_E) of C^*(E) are outer but not inner conjugate. For the Cuntz algebras O_n, 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.