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2012 | 10 | 2 | 646-655
Tytuł artykułu

Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

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Języki publikacji
EN
Abstrakty
EN
It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $\mathbb{F}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where $\mathbb{F}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, $\mathbb{F}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
2
Strony
646-655
Opis fizyczny
Daty
wydano
2012-04-01
online
2012-01-18
Twórcy
autor
Bibliografia
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  • [9] Jarosz K., Function representation of a noncommutative uniform algebra, Proc. Amer. Math. Soc., 2008, 136(2), 605–611 http://dx.doi.org/10.1090/S0002-9939-07-09033-8
  • [10] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., 2010, 40(6), 1903–1922 http://dx.doi.org/10.1216/RMJ-2010-40-6-1903
  • [11] Kaniuth E., A Course in Commutative Banach Algebras, Grad. Texts in Math., 246, Springer, New York, 2009 http://dx.doi.org/10.1007/978-0-387-72476-8
  • [12] Kulkarni S.H., Limaye B.V., Real Function Algebras, Monogr. Textbooks Pure Appl. Math., 168, Marcel Dekker, New York, 1992
  • [13] Lambert S., Luttman A., Generalized strong boundary points and boundaries of families of continuous functions, Mediterr. J. Math. (in press), DOI: 10.1007/s00009-010-0105-5
  • [14] Rickart C.E., General Theory of Banach Algebras, The University Series in Higher Mathematics, Van Nostrand, Princeton, 1960
  • [15] Šilov G., On the extension of maximal ideals, Dokl. Akad. Nauk SSSR, 1940, 29, 83–84 (in Russian)
  • [16] Weaver N., Lipschitz Algebras, World Scientific, River Edge, 1999
  • [17] Yates R.B.J., Norm-Preserving Criteria for Uniform Algebra Isomorphisms, PhD thesis, University of Montana, 2009
  • [18] Żelazko W., Banach Algebras, Elsevier, Amsterdam-London-New York, 1973
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0133-9
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