### 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 language | English |
---|---|

Pages (from-to) | 1291-1325 |

Number of pages | 35 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 33 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2013 Oct |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*33*(5), 1291-1325. https://doi.org/10.1017/S0143385712000405

**Dynamical systems of type (m,n) and their C*-algebras.** / Ara, Pere; Exel, Ruy; Katsura, Takeshi.

Research output: Contribution to journal › Article

*Ergodic Theory and Dynamical Systems*, vol. 33, no. 5, pp. 1291-1325. https://doi.org/10.1017/S0143385712000405

}

TY - JOUR

T1 - Dynamical systems of type (m,n) and their C*-algebras

AU - Ara, Pere

AU - Exel, Ruy

AU - Katsura, Takeshi

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84883436685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883436685&partnerID=8YFLogxK

U2 - 10.1017/S0143385712000405

DO - 10.1017/S0143385712000405

M3 - Article

AN - SCOPUS:84883436685

VL - 33

SP - 1291

EP - 1325

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 5

ER -