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Semigroups of local homeomorphisms and interaction groups

Published online by Cambridge University Press:  01 December 2007

R. EXEL  and
J. RENAULT
R. EXEL
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil (email: [email protected])
J. RENAULT
Affiliation:
Départment de Mathématiques, Université d’Orléans, 45067 Orléans, France (email: [email protected])
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Abstract

Given a semigroup of surjective local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. We also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1737 - 1771
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Arzumanian, V. and Renault, J.. Examples of pseudogroups and their C*-algebras. Operator Algebras and Quantum Field Theory (Rome, 1996). International Press, Cambridge, MA, 1997, pp. 93–104.Google Scholar
[2]Deaconu, V.. Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347 (1995), 1779–1786.CrossRefGoogle Scholar
[3]Deaconu, V.. C*-algebras of commuting endomorphisms. Advances in Operator Algebras and Mathematical Physics (Theta Series in Advanced Mathematics, 5). Theta, Bucharest, 2005, pp. 47–55 [arXiv:math.OA/0406624].Google Scholar
[4]Exel, R.. Amenability for Fell bundles. J. Reine Angew. Math. 492 (1997), 41–73 [arXiv:funct-an/9604009].Google Scholar
[5]Exel, R.. A new look at the crossed-product of a C*-algebra by an endomorphism. Ergod. Th. & Dynam. Sys. 23 (2003), 1733–1750 [arXiv:math.OA/0012084].CrossRefGoogle Scholar
[6]Exel, R.. A new look at the crossed-product of a C*-algebra by a semigroup of endomorphisms. Ergod. Th. & Dynam. Sys. to appear [arXiv:math.OA/0511061].Google Scholar
[7]Exel, R. and Vershik, A.. C*-algebras of irreversible dynamical systems. Canad. J. Math. 58 (2006), 39–63 [arXiv:math.OA/0203185].CrossRefGoogle Scholar
[8]Hedlund, G. A.. Endormorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375.CrossRefGoogle Scholar
[9]Ledrappier, F.. Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A–B 287(7) (1978), A561–A563.Google Scholar
[10]Renault, J.. A Groupoid Approach to C*-algebras (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.CrossRefGoogle Scholar
[11]Yeend, T.. Groupoid models for the C*-algebras of topological higher-rank graphs. J. Operator Theory 57(1) (2007), 95–120.Google Scholar