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Dynamical systems of type $(m,n)$ and their $\mathrm {C}^*$ -algebras

  • PERE ARA (a1), RUY EXEL (a2) and TAKESHI KATSURA (a3)


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 $\mathbb {F}_{m+n}$ , 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.



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Dynamical systems of type $(m,n)$ and their $\mathrm {C}^*$ -algebras

  • PERE ARA (a1), RUY EXEL (a2) and TAKESHI KATSURA (a3)


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