Boundary Quotients of C^*-algebras of LCM Semigroups with Irreversible Algebraic Dynamical Systems

Abstract

The known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere are given. For every such minimal dynamical system (S^n,β) there is a Cantor minimal system (X,α) such that the corresponding product system (X × S^n,α×β) is minimal and the resulting crossed product C^*-algebraC(X × S^n ) ⋊_(α×β) Zis tracially approximately an interval algebra. This entails classification for such C^*-algebras.We study the saturation properties of several classes of C^*-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of σ-unital C^*- algebras; we extend their results by showing that some coronas of non-σ-unital C^*-algebras are countably degree-1 saturated. We study C^*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. We develop a general notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as irreversible analogues of the dynamical systems considered by Schmidt.