Dynamical systems of type and their -algebras

Abstract

AbstractGiven 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$\mathbb {F}_{m+n}$, 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$m,n\geq 2$, we prove that the partial action of$\mathbb {F}_{m+n}$is topologically free and that${\cal O}_{m,n}^r$satisfies property (SP) (small projections). We also show that${\cal O}_{m,n}^r$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.