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

Pere Ara, Ruy Exel, Takeshi Katsura

Research output: Contribution to journalArticle

12 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 language English 1291-1325 35 Ergodic Theory and Dynamical Systems 33 5 https://doi.org/10.1017/S0143385712000405 Published - 2013 Oct

### Fingerprint

Partial Action
Crossed Product
Algebra
C*-algebra
Dynamical systems
Dynamical system
Partial Isometry
Process Mean
Generating Set
Discrete Group
Homeomorphic
Free Group
Topological space
Bundle
Disjoint
Quotient
Union
Projection
Transform
Integer

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

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

In: Ergodic Theory and Dynamical Systems, Vol. 33, No. 5, 10.2013, p. 1291-1325.

Research output: Contribution to journalArticle

@article{2fe4faefe80e42ae89d6f39af43d320a,
title = "Dynamical systems of type (m,n) and their C*-algebras",
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.",
author = "Pere Ara and Ruy Exel and Takeshi Katsura",
year = "2013",
month = "10",
doi = "10.1017/S0143385712000405",
language = "English",
volume = "33",
pages = "1291--1325",
journal = "Ergodic Theory and Dynamical Systems",
issn = "0143-3857",
publisher = "Cambridge University Press",
number = "5",

}

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 -