Ordered groups and crossed products of AF-related C*-algebras

Abstract

Let C be the canonical diagonal MASA in an AF algebra A. We show that an automorphism &phis; of C extends to an approximately inner automorphism alpha of A iff p is Murray-von Neumann equivalent in A to &phis;( p) for every projection p ∈ C. We determine some classes of AF algebras for which the ordered K 0-groups of the two crossed products, arising from these dynamical systems, are isomorphic and we show that the entropies of &phis; and alpha are equal. Along the way, we obtain results about when K0( A ⋉aZ ) (equipped with natural order) is pre-order isomorphic with K0(A)/Im(id-- K0(alpha)) (equipped with the quotient order) for a large class of automorphisms alpha of the AF algebra A. I. Putnam has proved that C(X) ⋉fZ is a (simple) AT algebra when X is the Cantor set and &phis; is a minimal self-homeomorphism of X. We extend his results to non-commutative AF algebras that have the set of extremal tracial states homeomorphic with the Cantor set, and satisfy a certain lifting property. The proof is based (as in the commutative case) on constructions of intermediate AF subalgebras (associated to closed subsets of deltaeT (A)) of the crossed product and on the lifting. We study the structure of the crossed products by Z of algebras of type C(X) ⊗ B, where X is the Cantor set, B is a unital, simple, topological tracial rank zero or one C*-algebra and the automorphism is alphaT ⊗ beta, where T is a minimal homeomorphism of X and beta ∈ Aut0. Using the work of Matui and Lin we show that the crossed product algebra has still topological tracial rank zero or one. The last topic addressed in this thesis concerns the natural map: Asa → AffT(A), where T(A) is the simplex of tracial states of a unital C*-algebra A. Considering the natural orders on Asa and on AffT(A ), we found examples of C*-algebras for which the natural maps lifts positive functions to positive elements in A.