Abstract
The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions \(\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}\), is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map \(n\mapsto \lambda^{n^{k}}\), to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra W k , the norm closure of the algebra generated by the set of functions \(\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}\), will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.
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References
Auslander, L. and Hahn, F., Real functions coming from flows on compact spaces and concepts of almost periodicity, Trans. Amer. Math. Soc. 106 (1963), 415–426.
Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on Semigroups : Function Spaces, Compactifications, Representations, Wiley, New York, 1989.
Dales, H. G. and Lau, A. T. -M., The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177:836 (2005).
Dales, H. G., Lau, A. T. -M. and Strauss, D., Banach algebras on semigroups and their compactifications, Mem. Amer. Math. Soc. 205:966 (2010).
Ellis, R., A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272–281.
Furstenberg, H., Strict ergodicity and transformations of the torus, Amer. J. Math. 83 (1961), 573–601.
Furstenberg, H., The structure of distal flows, Amer. J. Math. 85 (1963), 477–515.
Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis, vol. 1, Springer, Berlin, 1979.
Isik, N., Pym, J. and Ülger, A., The second dual of the group algebra of a compact group, J. Lond. Math. Soc. 35 (1987), 135–148.
Jabbari, A., The Topological Centre of Some Semigroup Compactifications, Ph.D. Thesis, Ferdowsi University of Mashhad, Mashhad, 2008.
Jabbari, A. and Namioka, I., Ellis group and the topological center of the flow generated by the map \(n\mapsto \lambda^{n^{k}}\), to appear in Milan J. Math.
Jabbari, A. and Vishki, H. R. E., A class of distal functions on semitopological semigroups, Methods Funct. Anal. Topology 15 (2009), 188–194.
Jabbari, A. and Vishki, H. R. E., Skew-product dynamical systems, Ellis groups and topological centre, Bull. Aust. Math. Soc. 79 (2009), 129–145.
Knapp, A. W., Distal functions on groups, Trans. Amer. Math. Soc. 128 (1967), 1–40.
Lau, A. T.-M., Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cabridge Philos. Soc. 99 (1986), 273–283.
Lau, A. T.-M. and Losert, V., On the second conjugate algebra of L 1(G) of a locally compact group, J. Lond. Math. Soc. 37 (1988), 464–470.
Lau, A. T.-M. and Loy, R. J., Banach algebras on compact right topological groups, J. Funct. Anal. 225 (2005), 293–300.
Lau, A. T.-M., Milnes, P. and Pym, J. S., Locally compact groups, invariant means and the centres of compactifications, J. Lond. Math. Soc. 56 (1997), 77–90.
Lau, A. T.-M. and Pym, J. S., The topological centre of a compactification of a locally compact group, Math. Z. 219 (1995), 567–579.
Milnes, P. and Pym, J., Haar measure for compact right topological groups, Proc. Amer. Math. Soc. 114 (1992), 387–393.
Milnes, P. and Pym, J., Homomorphism of minimal and distal flows, J. Nigerian Math. Soc. 11 (1992), 63–80.
Namioka, I., Affine flows and distal points, Math. Z. 184 (1983), 259–269.
Namioka, I., Ellis groups and compact right topological groups, in Conference in Modern Analysis and Probability (New Haven, CT , 1982 ), Contemp. Math. 26, pp. 295–300, Amer. Math. Soc., Providence, RI, 1984.
Salehi, E., Distal functions and unique ergodicity, Trans. Amer. Math. Soc. 323 (1991), 703–713.
Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.
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Jabbari, A. The topological center of the spectrum of some distal algebras. Ark Mat 50, 291–304 (2012). https://doi.org/10.1007/s11512-010-0132-2
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DOI: https://doi.org/10.1007/s11512-010-0132-2