Coverings of solenoids and automorphisms of semigroup $C^*$-algebras

R. N. Gumerov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 275-286

Résumé

The paper deals with finite-sheeted covering mappings onto the $P$-adic solenoids and limit endomorphisms of semigroup $C^*$-algebras. The aim of our exposition is two-fold: firstly, to present the results concerning the above-mentioned mappings and endomorphisms; secondly, to demonstrate proofs for some of the results. It has been shown that every covering mapping onto a solenoid is isomorphic to a power mapping. We have considered dynamical properties of the covering mappings. A power mapping for the $P$-adic solenoid is topologically transitive. A criterion for the covering mapping to be chaotic has been given. The classical Euler–Fermat theorem may be used in its proof. We have studied limit endomorphisms of $C^*$-algebras generated by isometric representations for semigroups of rational numbers. We formulate criteria for limit endomorphisms to be automorphisms in number-theoretic, algebraic, and functional terms. The necessity of such a criterion has been given from the category-theoretic viewpoint.

Détail

Keywords: automorphism of $C^*$-algebras, chaotic, inductive sequence of Toeplitz algebras associated with sequence of prime numbers, inverse limit and sequence, finite-sheeted covering mapping, semigroup $C^*$-algebra, Toeplitz algebra, topologically transitive.
Mots-clés : solenoid, $*$-homomorphism

Bibliographie

[1] Vietoris L., “Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen”, Math. Ann., 97 (1927), 454–472 | DOI | MR | Zbl

[2] Dantzig van D., Waerden van der B. L., “Über metrisch homogene Räume”, Abh. Math. Semin. Hamburg, 1928, no. 6, 367–376 | DOI | MR | Zbl

[3] Dantzig van D., “Über topologisch homogene Kontinua”, Fundam. Math., 1930, no. 15, 102–125 | DOI | Zbl

[4] Pontryagin L. S., Continuous Groups, Nauka, M., 1984, 520 pp. (In Russian) | MR

[5] Grigorian S. A., Gumerov R. N., “On a covering group theorem and its applications”, Lobachevskii J. Math., 10 (2002), 9–16 | MR | Zbl

[6] Grigorian S. A., Gumerov R. N., On the structure of finite coverings of compact connected groups, 2004, arXiv: math/0403329 | MR

[7] Grigorian S. A., Gumerov R. N., “On the structure of finite coverings of compact connected groups”, Topol. Its Appl., 153:18 (2006), 3598–3614 | DOI | MR | Zbl

[8] Gumerov R. N., “Weierstrass polynomials and coverings of compact groups”, Sib. Math. J., 54:2 (2013), 243–246 | DOI | MR | Zbl

[9] Gumerov R. N., “Characters and coverings of compact groups”, Russ. Math., 58:4 (2014), 7–13 | DOI | MR | Zbl

[10] Hewitt E., Ross K. A., Abstract Harmonic Analysis, v. I, Structure of Topological Groups Integration Theory Group Representations, Springer, Berlin, 1963, 525 pp. | DOI | MR | Zbl

[11] Keesling J., “The group of homeomorphisms of a solenoid”, Trans. Amer. Math. Soc., 172 (1972), 119–131 | DOI | MR

[12] Krupski P., “Means on solenoids”, Proc. Am. Math. Soc., 131:6 (2002), 1931–1933 | DOI | MR

[13] Gumerov R. N., “On the existence of means on solenoids”, Lobachevskii J. Math., 17 (2005), 43–46 | MR | Zbl

[14] Grigorian S. A. Gumerov R. N., Kazantsev A. V., “Group structure in finite coverings of compact solenoidal groups”, Lobachevskii J. Math., 4 (2000), 39–46 | MR

[15] Eda K., Matijević V., “Finite-sheeted covering maps over 2-dimensional connected, compact Abelian groups”, Topol. Its Appl., 153:7 (2006), 1033–1045 | DOI | MR | Zbl

[16] Eda K., Matijevic V., “Existence and uniqueness of group structures on covering spaces over groups”, Fundam. Math., 238 (2017), 241–267 | DOI | MR | Zbl

[17] Dydak J., “Overlays and group actions”, Topol. Its Appl., 207 (2016), 22–32 | DOI | MR | Zbl

[18] Fox R. H., “On shape”, Fundam. Math., 74 (1972), 47–71 | DOI | MR | Zbl

[19] Fox R. H., “Shape theory and covering spaces”, Lecture Notes in Mathematics, 375, eds. Dickman R. F., Fletcher P., Springer, Berlin–Heidelberg, 1974, 71–90 | DOI | MR

[20] Moore T. T., “On Fox's theory of overlays”, Fundam. Math., 99 (1978), 205–211 | DOI | MR | Zbl

[21] Zhou Youcheng, “Covering mapping on solenoids and their dynamical properties”, Chin. Sci. Bull., 45:12 (2000), 1066–1070 | DOI | MR | Zbl

[22] Charatonik J. J., Covarrubias P. P., “On covering mappings on solenoids”, Proc. Am. Math. Soc., 130:7 (2002), 2145–2154 | DOI | MR | Zbl

[23] Bogatyi S. A., Frolkina O. D., “Classification of generalized solenoids”, Proc. Semin. on Vector and Tensor Analysis, XXVI, Mosk. Gos. Univ., M., 2005, 31–59 (In Russian)

[24] Gumerov R. N., “On finite-sheeted covering mappings onto solenoids”, Proc. Am. Math. Soc., 133:9 (2005), 2771–2778 | DOI | MR | Zbl

[25] Brownlowe N., Raeburn I., “Two families of Exel-Larsen crossed products”, J. Math. Anal. Appl., 398:1 (2013), 68–79 | DOI | MR | Zbl

[26] Gumerov R. N., “Dynamical properties of covering mappings of solenoids”, Proc. Int. Conf. “Function Spaces, Approximation Theory, Nonlinear Analysis” dedicated to the centennial of S. M. Nikolskii (Moscow, Russia, May 23–29, 2005), M., 2005, 92 (In Russian)

[27] Eda K., Matijevic V., “Covering maps over solenoids which are not covering homomorphisms”, Fundam. Math., 221 (2013), 69–82 | DOI | MR | Zbl

[28] Coburn L. A., “The $C^*$-algebra generated by an isometry”, Bull. Am. Math. Soc., 73:5 (1967), 722–726 | DOI | MR | Zbl

[29] Douglas R. G., “On the $C^*$-algebra of a one-parameter semigroup ofisometries”, Acta Math., 128 (1972), 143–152 | DOI | MR

[30] Murphy G. J., “Ordered groups and Toeplitz algebras”, J. Oper. Theory, 18:2 (1987), 303–326 | MR | Zbl

[31] Grigoryan S. A., Salakhutdinov A. F., “$C^*$-algebras generated by cancellative semigroups”, Sib. Math. J., 51:1 (2010), 12–19 | DOI | MR | Zbl

[32] Grigoryan T. A., Lipaheva E. V., Tepoyan V. A., “On the extension of the Toeplitz algebra”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 154:4 (2012), 130–138 (In Russian) | MR

[33] Li X., “Semigroup $C^*$-algebras and amenability of semigroups”, J. Funct. Anal., 262:10 (2012), 4302–4340 | DOI | MR | Zbl

[34] Lipacheva E. V., Hovsepyan K. H., “Automorphisms of some subalgebras of the Toeplitz algebra”, Sib. Math. J., 57:3 (2016), 525–531 | DOI | MR | Zbl

[35] 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., 122:4 (1994), 1133–1141 | DOI | MR | Zbl

[36] Murphy G. J., $C^*$-algebras and operator theory, Academic Press, New York, 1990, 286 pp. | MR

[37] Gumerov R. N., “Limit automorphisms of $C^*$-algebras generated by isometric representations for semigroups of rational numbers”, Sib. Math. J., 59:1 (2018), 73–84 | DOI | MR | Zbl

[38] Helemskii A. Ya., Banach and Locally Convex Algebras, Clarendon Press, Oxford Univ. Press, New York, 1993, 464 pp. | MR