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Limit automorphisms of the \(C^\ast\)-algebras generated by isometric representations for semigroups of rationals. (English. Russian original) Zbl 1401.46037

Sib. Math. J. 59, No. 1, 73-84 (2018); translation from Sib. Mat. Zh. 59, No. 1, 95-109 (2018).
Summary: We consider inductive sequences of Toeplitz algebras whose connecting homomorphisms are defined by collections of primes. The inductive limits of these sequences are \(C^\ast\)-algebras generated by representations for semigroups of rationals. We study the limit endomorphisms of these \(C^\ast\)-algebras induced by morphisms between copies of the same inductive sequences of Toeplitz algebras. We establish necessary and sufficient conditions for these endomorphisms to be automorphisms of the algebras.

MSC:

46L05 General theory of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
46M10 Projective and injective objects in functional analysis
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
Full Text: DOI

References:

[1] Helemskiĭ A. Ya., Banach and Normed Algebras: General Theory, Representations, Homology [Russian], Nauka, Moscow (1989). · Zbl 0688.46025
[2] Murphy G., C* -Algebras and Operator Theory, Academic Press, Boston (1990). · Zbl 0714.46041
[3] Coburn, L. A., The \(C\)* -algebra generated by an isometry, Bull. Amer. Math. Soc., 73, 722-726, (1967) · Zbl 0153.16603 · doi:10.1090/S0002-9904-1967-11845-7
[4] Coburn, L. A., The \(C\)* -algebra generated by an isometry. II, Trans. Amer. Math. Soc., 137, 211-217, (1969) · Zbl 0186.19704
[5] Douglas, R. G., On the \(C\)* -algebra of a one-parameter semigroup of isometries, Acta Math., 128, 143-152, (1972) · Zbl 0232.46063 · doi:10.1007/BF02392163
[6] Murphy, G. J., Ordered groups and Toeplitz algebras, J. Oper. Theory, 18, 303-326, (1987) · Zbl 0656.47020
[7] Murphy, G. J., Simple \(C\)* -algebras and subgroups of Q, Proc. Amer. Math. Soc., 107, 97-100, (1989) · Zbl 0682.46041
[8] Murphy, G. J., Toeplitz operators and algebras, Math. Z., 208, 355-362, (1991) · Zbl 0725.47026 · doi:10.1007/BF02571532
[9] Grigoryan, S. A.; Salakhutdinov, A. F., \(C\)* -algebras generated by cancellative semigroups, Sib. Math. J., 51, 12-19, (2010) · Zbl 1210.46047 · doi:10.1007/s11202-010-0002-y
[10] Grigoryan, T. A.; Lipacheva, E. V.; Tepoyan, V. A., On the extension of the Toeplitz algebra, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 154, 130-138, (2012) · Zbl 1285.47088
[11] Lipacheva, E. V.; Hovsepyan, K. H., Automorphisms of some subalgebras of the Toeplitz algebra, Sib. Math. J., 57, 525-531, (2016) · Zbl 1366.46050 · doi:10.1134/S0037446616030149
[12] Zhou, Y., Covering mappings on solenoids and their dynamical properties, Chinese Sci. Bull., 45, 1066-1070, (2000) · Zbl 1330.37021 · doi:10.1007/BF02887175
[13] Charatonik, J. J.; Covarrubias, P. P., On covering mappings on solenoids, Proc. Amer. Math. Soc., 130, 2145-2154, (2002) · Zbl 0989.54038 · doi:10.1090/S0002-9939-01-06296-7
[14] Grigorian, S. A.; Gumerov, R. N., On a covering group theorem and its applications, Lobachevskii J. Math., 10, 9-16, (2002) · Zbl 1010.22007
[15] Gumerov, R. N., On finite-sheeted covering mappings onto solenoids, Proc. Amer. Math. Soc., 133, 2771-2778, (2005) · Zbl 1075.54015 · doi:10.1090/S0002-9939-05-07792-0
[16] Bogatyi, S. A.; Frolkina, O. D., Classification of generalized solenoids, Proceedings of the Seminar on Vector and Tensor Analysis with Applications to Geometry, Mechanics, and Physics, XXVI, 31-59, (2005) · Zbl 1133.54318
[17] Grigorian, S. A.; Gumerov, R. N., On the structure of finite coverings of compact connected groups, Topology Appl., 153, 3598-3614, (2006) · Zbl 1110.57001 · doi:10.1016/j.topol.2006.03.010
[18] Gumerov, R. N., Weierstrass polynomials and coverings of compact groups, Sib. Math. J., 54, 243-246, (2013) · Zbl 1275.54024 · doi:10.1134/S0037446613020080
[19] Brownlowe, N.; Raeburn, I., Two families of exel-Larsen crossed products, J. Math. Anal. Appl., 398, 68-79, (2013) · Zbl 1270.46061 · doi:10.1016/j.jmaa.2012.08.026
[20] Keesling, J., The group of homeomorphisms of a solenoid, Trans. Amer. Math. Soc., 172, 119-131, (1972) · Zbl 0249.57022 · doi:10.1090/S0002-9947-1972-0315735-6
[21] Krupski, P., Means on solenoids, Proc. Amer. Math. Soc., 131, 1931-1933, (2002) · Zbl 1029.54041 · doi:10.1090/S0002-9939-02-06738-2
[22] Gumerov, R. N., On the existence of means on solenoids, Lobachevskii J. Math., 17, 43-46, (2005) · Zbl 1070.54017
[23] Li, X., Semigroup \(C\)* -algebras and amenability of semigroups, J. Funct. Anal., 262, 4302-4340, (2012) · Zbl 1243.22006 · doi:10.1016/j.jfa.2012.02.020
[24] Li, X., Nuclearity of semigroup \(C\)* -algebras and the connection to amenability, Adv. Math., 244, 626-662, (2013) · Zbl 1293.46030 · doi:10.1016/j.aim.2013.05.016
[25] Adji, S.; Laca, M.; Nilsen, M.; Raeburn, I., Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups, Proc. Amer. Math. Soc., 122, 1133-1141, (1994) · Zbl 0818.46071 · doi:10.1090/S0002-9939-1994-1215024-1
[26] Rordam M., Larsen F., and Lausten N., An Introduction to K-Theory for C* -Algebras, Cambridge Univ. Press, Cambridge (2000) (London Math. Soc. Student Texts; vol. 49). · Zbl 0794.54020
[27] Hewitt E. and Ross K., Abstract Harmonic Analysis. Vol. 1, Springer-Verlag, New York (1994). · Zbl 0837.43002
[28] Gumerov, R. N., Characters and coverings of compact groups, Russian Math. (Iz. VUZ), 58, 7-13, (2014) · Zbl 1304.22003 · doi:10.3103/S1066369X14040021
[29] Charatonik, J. J., Some problems concerning means on topological spaces, Topology, Measures and Fractals, 66, 166-177, (1992) · Zbl 0794.54020
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