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On Gelfand-Mazur theorem on a class ofF-algebras

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EN
A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence (xn) in A, (xn) is Cauchy whenever the sequence bn(xn − xn-1) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra, then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theorem for a large class of F-algebras, containing both locally bounded algebras and locally convex algebras with bounded elements.
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2
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1
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Daty
otrzymano
2013-04-11
zaakceptowano
2014-04-01
online
2014-08-08
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autor
Bibliografia
  • [1] M. Abel, Topological algebras with idempotently pseudoconvex von Neumann bornology, Contemp. Math. 427 (2007) 15–29.
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  • [3] G. R. Allan, A spectral theory for locally convex algebra, Proc. Landon Math. Soc. 115 (1965) 399–421.
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  • [5] R. Arens, Linear topological division algebras, Bull. Amer. Math. Soc. 53 (1947) 623–630.
  • [6] V. M. Bogdan, On Frobenius, Mazur, and Gelfand-Mazur Theorems on Division Algebras, Quaestiones Mathematicae 29(2006) 171–209.
  • [7] F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, Heidelberg and Berlin, 1973.
  • [8] A. Ya. Helemskii, Banach and locally convex algebras, Oxford university press, 1993.
  • [9] A. Mallios, Topological algebras, selected topics, Mathematical Studies, North Holland, Amsterdam, 1986.
  • [10] W. Zelazko, A theorem on B0 division algebras, Bull. Polon. Acad. Sc. 8 (1960) 373–375.
  • [11] W. Zelazko, F-algebras: Some results and open problems, Functional Analysis and its Applications 197 (2004) 317–326.
  • [12] W. Zelazko, On the locally bounded and m-convex topological algebras, Studia Math. 19 (1960) 333–356.
  • [13] W. Zelazko, continuous characters and joint topological spectrum, Control and Cybernetics 36 (2007) 859–864.
  • [14] W. Zelazko, Selected Topics in Topological Alghebras, Aarhus Univ. Lecture Notes 31 1971.
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_taa-2014-0004
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