Covariance algebra of a partial dynamical system

Bartosz Kosma Kwaśniewski

Open Mathematics, Tome 3 (2005), p. 718-765 / Harvested from The Polish Digital Mathematics Library

Résumé

A pair (X, α) is a partial dynamical system if X is a compact topological space and α: Δ→ X is a continuous mapping such that Δ is open. Additionally we assume here that Δ is closed and α(Δ) is open. Such systems arise naturally while dealing with commutative C *-dynamical systems. In this paper we construct and investigate a universal C *-algebra C *(X,α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto. The main method here is to use the description of maximal ideal space of a coefficient algebra, cf. [16, 18], in order to construct a larger system, (X˜,α˜) where α˜ is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13], In particular, one generalizes the notions of topological freeness and invariance of a set, which are being use afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C *(X, α).

Détail

Publié le : 2005-01-01

Zbl 1117.46046

EUDML-ID : urn:eudml:doc:268682

Bibliographie

[1] S. Adji, M. Laca, M. Nilsen and I. Raeburn: “Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups”, Proc. Amer. Math. Soc., Vol. 122(4), (1994), pp. 1133–1141. http://dx.doi.org/10.2307/2161182 | Zbl 0818.46071

[2] A. Antonevich: Linear Functional Equations. Operator Approach, Operator Theory Advances and Applications, Vol. 83, Birkhauser Verlag, Basel, 1996.

[3] A. Antonevich and A.V. Lebedev: Functional differential equations: I. C *-theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 70, Longman Scientific & Technical, Harlow, Essex, England, 1994. | Zbl 0799.34001

[4] B. Blackadar: K-theory for Operator algebras, Springer-Verlag, New York, 1986.

[5] T. Bates, D. Pask, I. Raeburn and W. Szymański: “C*-algebras of row-finite graphs”, New York J. Math. Vol. 6, (2000), pp. 307–324. | Zbl 0976.46041

[6] O. Bratteli and D. W. Robinson: Operator algebras and Quantum Statistical Mechanics I, II New York 1979, 1980.

[7] M. Brin and G. Stuck: Introduction to Dynamical Systems, Cambridge University Press, 2003.

[8] J. Cuntz: “Simple C *-algebras generated by isometries”, Commun. Math. Phys., Vol. 57, (1977), pp. 173–185. http://dx.doi.org/10.1007/BF01625776 | Zbl 0399.46045

[9] J. Dixmier: C *-algebres et leurs representations, Gauter-Villars, Paris, 1969.

[10] R. Exel: “Circle actions on C *-algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequence”, J. Funct. analysis, Vol. 122, (1994), pp. 361–401. http://dx.doi.org/10.1006/jfan.1994.1073 | Zbl 0808.46091

[11] R. Exel and M. Laca: “Cuntz-Krieger algebras for infinite matrices”, J. Reine Angew. Math., Vol. 512, (1999), pp. 119–172. http://front.math.ucdavis.edu/funct-an/9712008 | Zbl 0932.47053

[12] R. Exel: “A new look at the crossed-product of a C *-algebra by an endomorphism”, Ergodic Theory Dynam. Systems, Vol. 23, (2003), pp. 1733–1750, http://front.math.ucdavis.edu/math.OA/0012084 http://dx.doi.org/10.1017/S0143385702001797 | Zbl 1059.46050

[13] R. Exel, M. Laca and J. Quigg: “Partial dynamical systems and C *-algebras generated by partial isometries”, J. Operator Theory, Vol. 47, (2002), pp. 169–186. http://front.math.ucdavis.edu/funct-an/9712007. | Zbl 1029.46106

[14] R. Exel and D. Royer: “The crossed product by a partial endomorphism” Munuscript at arXiv:math.OA/0410192 vl 6 Oct 2004, http://front.math.ucdavis.edu/math.OA/0410192.

[15] R. Exel and A. Vershik: “C*-algebras, of irreversible dynamical systems”, To appear in Canadian J. Math., http://front.math.ucdavis.edu/math.OA/0203185. | Zbl 1104.46037

[16] B.K. Kwaśniewski and A.V. Lebedev: “Maximal ideal space of a commutative coefficient algebra”, preprint, http://front.math.ucdavis.edu/math.OA/0311416. | Zbl 1183.46065

[17] M. Laca and I. Raeburn: “A semigroup crossed product arising in number theory”, J. London Math. Soc., Vol. 59, (1999), pp. 330–344. http://dx.doi.org/10.1112/S0024610798006620 | Zbl 0922.46058

[18] A.V. Lebedev and A. Odzijewicz: “Extensions of C *-algebras by partial isometries”, Matem. Sbornik, Vol. 195(7), (2004), pp. 37–70. http://front.math.ucdavis.edu/math.OA/0209049

[19] A.V. Lebedev: “Topologically free partial actions and faithful representations of partial crossed products”, preprint, http://front.math.ucdavis.edu/math.OA/0305105. | Zbl 1122.46048

[20] K. McClanachan: “K-theory for partial crossed products by discrete groups”, J. Funct. Analysis, Vol. 130, (1995), pp. 77–117. http://dx.doi.org/10.1006/jfan.1995.1064 | Zbl 0867.46045

[21] G.J. Murphy: C *-algebras and operator theory, Academic Press, 1990.

[22] G.J. Murphy: “Crossed products of C *-algebras by endomorphisms”, Integral Equations Oper. Theory, Vol. 24, (1996), pp. 298–319. http://dx.doi.org/10.1007/BF01204603 | Zbl 0843.46050

[23] D.P. O’Donovan: “Weighted shifts and covariance algebras”, Trans. Amer. Math. Soc., Vol. 208, (1975), pp. 1–25. http://dx.doi.org/10.2307/1997272 | Zbl 0308.46050

[24] W.L. Paschke: “The Crossed Product of a C *-algebra by an Endomorphism”, Proc. Amer. math. Soc., Vol. 80, (1980), pp. 113–118. http://dx.doi.org/10.2307/2042155 | Zbl 0435.46045

[25] G.K. Pedersen: C *-algebras and their automorphism groups, Academic Press, London, 1979. | Zbl 0416.46043

[26] P.J. Stacey: “Crossed products of C *-algebras by *-endomorphisms”, J. Austral. Math. Soc., Vol. 54, (1993), pp. 204–212. http://dx.doi.org/10.1017/S1446788700037113 | Zbl 0785.46053

[27] R.F. Williams: “Expanding attractors”, IHES Publ. Math., Vol. 54, (1973), pp. 204–212. | Zbl 0279.58013