The Rokhlin property and the tracial topological rank

Abstract

Let A be a unital separable simple C ∗ -algebra with TR( A)⩽1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then TR(A⋊ α Z)⩽1 . We also show that whenever A has a unique tracial state and α m is uniformly outer for each m(≠0) and α r is approximately inner for some r>0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C ∗ -algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple A T -algebra of real rank zero and α∈Aut( A) which is approximately inner and if α satisfies some Rokhlin property, then the crossed product A⋊ α Z is again an A T -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C ∗ -algebras with tracial rank one (and zero) from crossed products.