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A new look at the crossed product of aC*-algebra by a semigroup of endomorphisms

Published online by Cambridge University Press:  01 June 2008

RUY EXEL
RUY EXEL*
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil (email: [email protected])
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Abstract

Let G be a group and let be a subsemigroup. In order to describe the crossed product of a C*-algebra A by an action of P by unital endomorphisms we find that we must extend the action to the whole group G. This extension fits into a broader notion of interaction groups which consists of an assignment of a positive operator Vg on A for each g in G, obeying a partial group law, and such that (Vg,Vg−1) is an interaction for every g, as defined in a previous paper by the author. We then develop a theory of crossed products by interaction groups and compare it to other endomorphism crossed product constructions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 749 - 789
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Bratteli, O. and Robinson, D. W.. Operator Algebras and Quantum Statistical Mechanics, 2 (Texts and Monographs in Physics). Springer, Berlin, 1997.CrossRefGoogle Scholar
[2]Brownlowe, N. and Raeburn, I.. Exel’s crossed product and relative Cuntz–Pimsner algebras. Math. Proc. Cambridge Philos. Soc. 141(3) (2006), 497–508 [arXiv:math.OA/0408324].CrossRefGoogle Scholar
[3]Cuntz, J.. Simple C *-algebras generated by isometries. Comm. Math. Phys. 57 (1977), 173–185.CrossRefGoogle Scholar
[4]Cuntz, J.. Automorphisms of certain simple C *-algebras. Quantum Fields—Algebras, Processes (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978). Springer, Vienna, 1980, pp. 187–196.CrossRefGoogle Scholar
[5]Exel, R.. Amenability for Fell bundles. J. Reine Angew. Math. 492 (1997), 41–73 [arXiv:funct-an/9604009], MR 99a:46131.Google Scholar
[6]Exel, R.. Partial actions of groups and actions of inverse semigroups. Proc. Amer. Math. Soc. 126 (1998), 3481–3494 [arXiv:funct-an/9511003], MR 99b:46102.CrossRefGoogle Scholar
[7]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
[8]Exel, R.. Interactions. J. Funct. Anal. 244(1) (2007), 26–62 [arXiv:math.OA/0409267].CrossRefGoogle Scholar
[9]Exel, R. and Renault, J.. Semigroups of local homeomorphisms and interaction groups. Ergod. Th. & Dynam. Sys. 27(6) (2007), 1737–1771 [arXiv:math.OA/0608589].CrossRefGoogle Scholar
[10]Exel, R. and Vershik, A.. C *-algebras of irreversible dynamical systems. Canad. J. Math. 58 (2006), 39–63 [arXiv:math.OA/0203185].CrossRefGoogle Scholar
[11]Fell, J. M. G. and Doran, R. S.. Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles (Pure and Applied Mathematics, 125 and 126). Academic Press, Boston, 1988.Google Scholar
[12]Fowler, N. J.. Discrete product systems of Hilbert bimodules. Pacific J. Math. 204 (2002), 335–375.CrossRefGoogle Scholar
[13]Katsura, T.. A construction of C *-algebras from C *-correspondences. Advances in Quantum Dynamics (South Hadley, MA, 2002) (Contemporary Mathematics). American Mathematical Society, Providence, RI, 2003, pp. 173–182 [arXiv:math.OA/0309059].CrossRefGoogle Scholar
[14]Kirchberg, E. and Wassermann, S.. Operations on continuous bundles of C *-algebras. Math. Ann. 303(4) (1995), 677–697.CrossRefGoogle Scholar
[15]Laca, M. and Raeburn, I.. Semigroup crossed products and Toeplitz algebras of non-abelian groups. J. Funct. Anal. 139 (1996), 415–440.CrossRefGoogle Scholar
[16]Larsen, N. S.. Crossed products by abelian semigroups via transfer operators. Preprint, 2005 [arXiv:math.OA/0502307].Google Scholar
[17]Pimsner, M. V.. A class of C *-algebras generalizing both Cuntz–Krieger algebras and crossed products by . Fields Inst. Commun. 12 (1997), 189–212.Google Scholar
[18]Renault, J.. A Groupoid Approach to C *-algebras (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.CrossRefGoogle Scholar
[19]Takesaki, M.. Theory of Operator Algebras, I. Springer, Berlin, 1979.CrossRefGoogle Scholar