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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)

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 $\mathbb {F}_{m+n}$ , 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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[AA]Abrams, G. and Aranda Pino, G.. The Leavitt path algebra of a graph. J. Algebra 293 (2005), 319334.
[AG1]Ara, P. and Goodearl, K. R.. Leavitt path algebras of separated graphs J. Reine Angew. Math. doi:10.1515/CRELLE.2011.146 [arXiv:1004.4979v2 math.RA].
[AG2]Ara, P. and Goodearl, K. R.. C*-algebras of separated graphs. J. Funct. Anal. 261 (2011), 25402568.
[AMP]Ara, P., Moreno, M. A. and Pardo, E.. Nonstable $K$-theory for graph algebras. Algebr. Represent. Theory 10 (2007), 157178.
[B]Brown, L. G.. Ext of certain free product C*-algebras. J. Operator Theory 6 (1981), 135141.
[BO]Brown, N. P. and Ozawa, N.. C*-Algebras and Finite-dimensional Approximations (Graduate Studies in Mathematics, 88). American Mathematical Society, Providence, RI, 2008.
[C]Cohen, D. E.. Combinatorial Group Theory: A Topological Approach (London Mathematical Society Student Texts, 14). Cambridge University Press, Cambridge, 1989.
[CK]Cuntz, J. and Krieger, W.. A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), 251268.
[DP]Duncan, J. and Paterson, A. L. T.. C*-algebras of inverse semigroups. Proc. Edinb. Math. Soc. (2) 28(1) (1985), 4158.
[E1]Exel, R.. Circle actions on C*-algebras, partial automorphisms and a generalized Pimsner–Voiculescu exact sequence. J. Funct. Anal. 122 (1994), 361401.
[E2]Exel, R.. Amenability for Fell bundles. J. Reine Angew. Math. 492 (1997), 4173.
[E3]Exel, R.. Partial representations and amenable Fell bundles over free groups. Pacific J. Math. 192 (2000), 3963.
[E4]Exel, R.. Exact groups, induced ideals, and Fell bundles. Preprint, 2003; arXiv:math.OA/0012091 (this paper was published as ‘Exact groups and Fell bundles’, Math. Ann. 323 (2002), (2), 259–266. However, the referee required that the results pertaining to induced ideals be removed from the preprint version arguing that there were no applications of this concept. The reader will therefore have to consult the arxiv version, where the results we need may be found).
[E5]Exel, R.. Twisted partial actions: a classification of regular $\mathrm {C}^*$-algebraic bundles. Proc. Lond. Math. Soc. (3) 74 (1997), 417443.
[EL]Exel, R. and Laca, M.. Cuntz–Krieger algebras for infinite matrices. J. Reine Angew. Math. 512 (1999), 119172.
[ELQ]Exel, R., Laca, M. and Quigg, J.. Partial dynamical systems and C*-algebras generated by partial isometries. J. Operator Theory 47(1) (2002), 169186; arXiv:funct-an/9712007v1.
[FD]Fell, J. M. G. and Doran, R. S.. Representations of *-algebras, Locally Compact Groups, and Banach *-algebraic Bundles. Vols. 1 & 2 (Pure and Applied Mathematics, 126). Academic Press, 1988.
[H]Hazrat, R.. The graded structure of Leavitt path algebras. Israel J. Math. to appear [arXiv:1005.1900v6  math.RA].
[KR]Kirchberg, E. and Rørdam, M.. Infinite non-simple $C^*$-algebras: Absorving the Cuntz algebra ${\cal O}_{\infty }$. Adv. Math. 167 (2002), 195264.
[L]Leavitt, W. G.. The module type of a ring. Trans. Amer. Math. Soc. 103 (1962), 113130.
[M1]McClanahan, K.. C*-algebras generated by elements of a unitary matrix. J. Funct. Anal. 107 (1992), 439457.
[M2]McClanahan, K.. K-theory and Ext-theory for rectangular unitary C*-algebras. Rocky Mountain J. Math. 23 (1993), 10631080.
[M3]McClanahan, K.. Simplicity of reduced amalgamated products of C*-algebras. Canad. J. Math. 46 (1994), 793807.
[M4]McClanahan, K.. $K$-theory for partial crossed products by discrete groups. J. Funct. Anal. 130 (1995), 77117.

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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