Dynamical systems of type (m,n) and their C*-algebras

TL;DR

This paper explores the structure and properties of C*-algebras arising from (m,n)-dynamical systems, revealing their non-exactness, topological freeness, and the absence of finite-dimensional representations, with new insights into Fell bundles.

Contribution

It introduces a new class of C*-algebras from (m,n)-dynamical systems, analyzing their exactness, topological properties, and representation theory, along with new results on Fell bundles.

Findings

O_{mn} is not exact for m,n ≥ 2

O_{mn}^r is exact and non-nuclear

O_{mn}^r admits no finite-dimensional representations

Abstract

Given positive integers n and m, we consider dynamical systems in which n copies of a topological space is homeomorphic to m copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra we denote by O_{mn}, which in turn is obtained as a quotient of the well known Leavitt C*-algebra L_{mn}, a process meant to transform the generating set of partial isometries of L{mn} into a tame set. Describing O_{mn} as the crossed-product of the universal (m,n)-dynamical system by a partial action of the free group F_{m+n}, we show that O_{mn} is not exact when n and m are both greater than or equal to 2, but the corresponding reduced crossed-product, denoted O_{mn}^r, is shown to be exact and non-nuclear. Still under the assumption that m,n>=2, we prove that the partial action of F_{m+n} is topologically free and that O_{mn}^r satisfies property (SP) (small projections). We also show that O_{mn}^r admits no finite dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.