Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

Abstract

We study the existence of minimal dynamical systems, their orbit and minimal orbit-breaking equivalence relations, and their applications to C * -algebras and K-theory. We show that given any finite CW-complex there exists a space with the same K-theory and cohomology that admits a minimal homeomorphism. The proof relies on the existence of homeomorphisms on point-like spaces constructed by the authors in previous work, together with existence results for skew product systems due to Glasner and Weiss. To any minimal dynamical system one can associate minimal equivalence relations by breaking orbits at small subsets. Using Renault's groupoid C * -algebra construction we can associate K-theory groups to minimal dynamical systems and orbit-breaking equivalence relations. We show that given arbitrary countable abelian groups G 0 and G 1 we can find a minimal orbit-breaking relation such that the K-theory of the associated C * -algebra is exactly this pair. These results have important applications to the Elliott classification program for C * -algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C * -algebras associated to minimal dynamical systems with mean dimension zero and their minimal orbit-breaking relations.

Constructing Menger manifold C*-diagonals in classifiable C*-algebras

“…Together with groupoid models (and hence Cartan subalgebras) which have already been constructed in the purely infinite case (see [55] and also [43, § 5]), this produces Cartan subalgebras in all classifiable C*-algebras. An alternative approach to constructing groupoid models, based on topological dynamics, has been developed in [10,52,11,12] and covers large classes of classifiable C*-algebras. In special cases, groupoid models have also been constructed in [3].The goal of this paper is to start a more detailed analysis of the C*-diagonals and the corresponding groupoids constructed in [42].…”

“…It suffices to show that for all y, ȳ ∈ X ⊆ Y q 0 that [0, y] ∼ conn [0, ȳ]. We further reduce to y, ȳ ∈ Y q conn : If y ∈ X [e Z ], then there exists r ∈ {0, 1} and ỹ ∈ Y r with b r ( ỹ) = b 0 (y) (= y), and by (12), we must have…”