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

Pere Ara, Ruy Exel, Takeshi Katsura

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

Given positive integers n and m, we consider dynamical systems in which (the disjoint union of) 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 denoted by {\cal O}-{m,n}, which in turn is obtained as a quotient of the well-known Leavitt C*-algebra L-{m,n}, a process meant to transform the generating set of partial isometries of L-{m,n} into a tame set. Describing {\cal O}-{m,n} 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 {\cal O}-{m,n} is not exact when n and m are both greater than or equal to 2, but the corresponding reduced crossed product, denoted by {\cal O}-{m,n} r, is shown to be exact and non-nuclear. Still under the assumption that m,n 2, we prove that the partial action of \mathbb F-m+n is topologically free and that O-m,n r satisfies property (SP) (small projections). We also show that {\cal O}-{m,n}^radmits 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.

Original languageEnglish
Pages (from-to)1291-1325
Number of pages35
JournalErgodic Theory and Dynamical Systems
Volume33
Issue number5
DOIs
Publication statusPublished - 2013 Oct

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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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