×

On the Exel crossed product of topological covering maps. (English) Zbl 1255.46032

Summary: For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product \(C^{*}\)-algebras \(C(X)\rtimes_{\alpha ,\mathcal L}\mathbb N\) introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of \(C(X)\) into \(C(X)\rtimes_{\alpha ,\mathcal L}\mathbb N\) is a maximal abelian \(C^{*}\)-subalgebra of \(C(X)\rtimes_{\alpha ,\mathcal L}\mathbb N\); any nontrivial two sided ideal of \(C(X)\rtimes_{\alpha ,\mathcal L}\mathbb N\) has non-zero intersection with the embedded copy of \(C(X)\); a certain natural representation of \(C(X)\rtimes_{\alpha ,\mathcal L}\mathbb N\) is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product \(C^{*}\)-algebras of homeomorphism dynamical systems.

MSC:

46L55 Noncommutative dynamical systems
47L65 Crossed product algebras (analytic crossed products)
47L40 Limit algebras, subalgebras of \(C^*\)-algebras
54H20 Topological dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H15 Transformation groups and semigroups (topological aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adji, S., Laca, M., Nilsen, M., Raeburn, I.: Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups. Proc. Am. Math. Soc. 122, 1133–1141 (1994) · Zbl 0818.46071 · doi:10.1090/S0002-9939-1994-1215024-1
[2] an Huef, A., Raeburn, I.: The ideal structure of Cuntz-Krieger algebras. Ergod. Theory Dyn. Syst. 17(3), 611–624 (1997) · Zbl 0886.46061 · doi:10.1017/S0143385797079200
[3] Archbold, R.J., Spielberg, J.S.: Topologically free actions and ideals in discrete C *-dynamical systems. Proc. Edinb. Math. Soc. 37, 119–124 (1993) · Zbl 0799.46076 · doi:10.1017/S0013091500018733
[4] Arzumanian, V.A., Vershik, A.M.: Factor representations of the crossed product of a commutative C*-algebra and a semigroup of its endomorphisms. Dokl. Akad. Nauk. SSSR 238, 513–517 (1978). Translated in Sov. Math. Dokl. 19(1) (1978)
[5] Arzumanian, V.A., Vershik, A.M.: Star algebras associated with endomorphisms. In: Operator Algebras and Group Representations. Proc. Int. Conf., Neptun/Rom. 1980, vol. I. Monogr. Stud. Math. 17, 17–27 (1984)
[6] Bratteli, O., Evans, D.E., Jorgensen, P.E.T.: Compactly supported wavelets and representations of the Cuntz relations. Appl. Comput. Harmon. Anal. 8(2), 166–196 (2000). · Zbl 0960.42013 · doi:10.1006/acha.2000.0283
[7] Bratteli, O., Jorgensen, P.E.T.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Am. Math. Soc. 139(663), 1–89 (1999). · Zbl 0935.46057
[8] Bratteli, O., Jorgensen, P.: Wavelets through a Looking Glass. The World of the Spectrum. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2002). xxii+398 pp. · Zbl 1012.42023
[9] Carlsen, T.M., Silvestrov, S.: C *-crossed products and shift spaces. Expo. Math. 25(4), 275–307 (2007) · Zbl 1154.46039
[10] Cuntz, J., Krieger, W.: A class of C *-algebras and topological Markov chains. Invent. Math. 56(3), 251–268 (1980) · Zbl 0434.46045 · doi:10.1007/BF01390048
[11] Dai, X., Larson, D.R.: Wandering vectors for unitary systems and orthogonal wavelets. Mem. Am. Math. Soc. 134(640), 68 (1998) · Zbl 0990.42022
[12] Davidson, K.R.: C *-Algebras by Example. Fields Institute Monographs. AMS, Providence (1996)
[13] Deaconu, V.: Groupoids associated with endomorphisms. Trans. Am. Math. Soc. 347(5), 1779–1786 (1995) · Zbl 0826.46058 · doi:10.2307/2154972
[14] Dutkay, D., Jorgensen, P.E.T.: Wavelet constructions in non-linear dynamics. Electron. Res. Announc. Am. Math. Soc. 11, 21–33 (2005) · Zbl 1069.42021 · doi:10.1090/S1079-6762-05-00143-5
[15] Dutkay, D.E., Jorgensen, P.E.T.: Hilbert spaces of martingales supporting certain substitution-dynamical systems. Conform. Geom. Dyn. 9, 24–45 (2005) · Zbl 1128.37005 · doi:10.1090/S1088-4173-05-00135-9
[16] Dutkay, D., Jorgensen, P.E.T.: Methods from multiscale theory and wavelets applied to nonlinear dynamics. In: Wavelets, Multiscale Systems and Hypercomplex Analysis. Oper. Theory Adv. Appl., vol. 167, pp. 87–126. Birkhäuser, Basel (2006) · Zbl 1148.37014
[17] Dutkay, D.E., Jorgensen, P.E.T.: Martingales, endomorphisms, and covariant systems of operators in Hilbert space. J. Oper. Theory 58(2), 269–310 (2007) · Zbl 1134.47305
[18] Effros, E.G., Hahn, F.: Locally compact transformation groups and C *-algebras. Mem. Am. Math. Soc. 75, 1–92 (1967) · Zbl 0184.17002
[19] Eilers, S.: C *-algebras associated to dynamical systems. Discrete Contin. Dyn. Syst. 15(1), 177–192 (2006) · Zbl 1104.46036 · doi:10.3934/dcds.2006.15.177
[20] Elliott, G.A.: Some simple C *-algebras constructed as crossed products with discrete outer automorphism groups. Publ. Res. Inst. Math. Sci. 16, 299–311 (1980) · Zbl 0438.46044 · doi:10.2977/prims/1195187509
[21] Exel, R.: Crossed-products by finite index endomorphisms and KMS states. J. Funct. Anal. 199(1), 153–188 (2003) · Zbl 1034.46056 · doi:10.1016/S0022-1236(02)00023-X
[22] Exel, R.: A new look at the crossed-product of a C *-algebra by an endomorphism. Ergod. Theory Dyn. Syst. 23(1), 1733–1750 (2003) · Zbl 1059.46050 · doi:10.1017/S0143385702001797
[23] Exel, R., Vershik, A.: C *-algebras of irreversible dynamical systems. Can. J. Math. 58(1), 39–63 (2006) · Zbl 1104.46037 · doi:10.4153/CJM-2006-003-x
[24] Jorgensen, P.E.T.: Operators and Representation Theory. North-Holland, Amsterdam (1988)
[25] Jorgensen, P.E.T.: Analysis and Probability: Wavelets, Signals, Fractals. Graduate Texts in Mathematics, vol. 234. Springer, New York (2006). xlviii+276 pp. · Zbl 1104.42001
[26] Kajiwara, T., Watatani, Y.: C *-algebras associated with complex dynamical systems. Indiana Univ. Math. J. 54(3), 755–778 (2005) · Zbl 1082.46045 · doi:10.1512/iumj.2005.54.2530
[27] Katsura, T.: A class of C *-algebras generalizing both graph algebras and homeomorphism C *-algebras. II. Examples. Int. J. Math. 17(7), 791–833 (2006) · Zbl 1107.46040 · doi:10.1142/S0129167X06003722
[28] Katsura, T.: A class of C *-algebras generalizing both graph algebras and homeomorphism C *-algebras. III. Ideal structures. Ergod. Theory Dyn. Syst. 26(6), 1805–1854 (2006) · Zbl 1136.46041 · doi:10.1017/S0143385706000320
[29] Kawamura, S.: Covariant representations associated with chaotic dynamical systems. Tokyo J. Math. 20(1), 205–217 (1997) · Zbl 0879.46034 · doi:10.3836/tjm/1270042409
[30] Kawamura, S., Tomiyama, J.: Properties of topological dynamical systems and corresponding C *-algebras. Tokyo J. Math. 13, 251–257 (1990) · Zbl 0724.54037 · doi:10.3836/tjm/1270132260
[31] Kishimoto, A.: Outer automorphisms and reduced crossed products of simple C *-algebras. Commun. Math. Phys. 81, 429–435 (1981) · Zbl 0467.46050 · doi:10.1007/BF01209077
[32] Mackey, G.W.: Induced Representations of Groups and Quantum Mechanics. W.A. Benjamin, New York, Editore Boringhieri, Torino (1968) · Zbl 0174.28101
[33] Mackey, G.W.: The Theory of Unitary Group Representations. The University of Chicago Press, Chicago (1976) · Zbl 0344.22002
[34] Mackey, G.W.: Unitary Group Representations in Physics, Probability, and Number Theory. Addison-Wesley, Reading (1989) · Zbl 0698.22001
[35] Matsumoto, K.: On C *-algebras associated with subshifts. Int. J. Math. 8(3), 357–374 (1997) · Zbl 0885.46048 · doi:10.1142/S0129167X97000172
[36] Ostrovskyĭ, V.L., Samoĭlenko, Yu.S.: Introduction to the Theory of Representations of Finitely Presented *-Algebras. I. Representations by Bounded Operators. Rev. Math. Phys. 11. The Gordon and Breach Publ. Group (1999)
[37] Pedersen, G.K.: C *-Algebras and their Automorphism Groups. Academic, New York (1979) · Zbl 0416.46043
[38] Quigg, J.C., Spielberg, J.S.: Regularity and hyporegularity in C *-dynamical system. Houst. J. Math. 18, 139–152 (1992) · Zbl 0785.46052
[39] Renault, J.: A Groupoid Approach to C*-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, New York (1980) · Zbl 0433.46049
[40] Renault, J.: Cuntz-like algebras. In: Operator Theoretical Methods (Timişoara, 1998), pp. 371–386. Theta Found., Bucharest (2000) · Zbl 1032.46535
[41] Renault, J.: Cartan subalgebras in C *-algebras. Irish Math. Soc. Bull. 61, 29–63 (2008) · Zbl 1175.46050
[42] Spielberg, J.S.: Free-product groups. Cuntz-Krieger algebras, and covariant maps. Int. J. Math. 2, 457–476 (1991) · Zbl 0769.46044 · doi:10.1142/S0129167X91000260
[43] Svensson, C., Tomiyama, J.: On the commutant of C(X) in C *-crossed products by \(\mathbb{Z}\) and their representations. arXiv:0807.2940 · Zbl 1178.46068
[44] Svensson, C., Silvestrov, S., de Jeu, M.: Dynamical systems and commutants in crossed products. Int. J. Math. 18, 455–471 (2007) · Zbl 1188.46041 · doi:10.1142/S0129167X07004217
[45] Svensson, C., Silvestrov, S., de Jeu, M.: Connections between dynamical systems and crossed products of Banach algebras by \(\mathbb{Z}\). In: Proceedings of Operator Theory, Analysis and Mathematical Physics, OTAMP-2006, Lund, Sweden, June 15–22, 2006 (to appear). (Preprints in Mathematical Sciences, Centre for Mathematical Sciences, Lund University 2007:5, LUTFMA-5081-2007; Leiden Mathematical Institute report 2007-02; arXiv:math/0702118 )
[46] Svensson, C., Silvestrov, S., de Jeu, M.: Dynamical systems associated with crossed products. In: Proceedings of Operator Methods in Fractal Analysis, Wavelets and Dynamical Systems, BIRS, Banff, Canada, December 2–December 7, 2006 (to appear). (Preprints in Mathematical Sciences 2007:22, LUTFMA-5088-2007; Leiden Mathematical Institute report 2007-30; arXiv:0707.1881 )
[47] Tomiyama, J.: The Interplay Between Topological Dynamics and Theory of C *-Algebras. Lecture Notes Series, vol. 2. Global Anal. Research Center, Seoul (1992) · Zbl 0892.46070
[48] Watatani, Y.: Index for C*-subalgebras. Mem. Am. Math. Soc. 424, 117 (1990) · Zbl 0697.46024
[49] Williams, D.P.: Crossed Products of C *-Algebras. Mathematical Surveys and Monographs, vol. 134. American Mathematical Society, Providence (2007). xvi+528 pp. · Zbl 1119.46002
[50] Zeller-Meier, G.: Produits croisés d’une C *-algèbre par un groupe d’automorphismes. J. Math. Pures Appl. 47, 101–239 (1968) · Zbl 0165.48403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.