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## Open Mathematics

2012 | 10 | 2 | 646-655
Tytuł artykułu

### Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

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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.
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EN
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Tom
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646-655
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wydano
2012-04-01
online
2012-01-18
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Bibliografia
• [1] Abel M., Jarosz K., Noncommutative uniform algebras, Studia Math., 2004, 162(3), 213–218 http://dx.doi.org/10.4064/sm162-3-2
• [2] Albiac F., Briem E., Representations of real Banach algebras, J. Aust. Math. Soc., 2010, 88(3), 289–300 http://dx.doi.org/10.1017/S144678871000011X
• [3] Albiac F., Briem E., Real Banach algebras as C(K) algebras, Q. J. Math. (in press), DOI: 10.1093/qmath/har005
• [4] Bear, H.S., The Silov boundary for a linear space of continuous functions, Amer. Math. Monthly, 1961, 68(5), 483–485 http://dx.doi.org/10.2307/2311109
• [5] Browder A., Introduction to Function Algebras, Math. Lecture Note Ser., W.A. Benjamin, New York-Amsterdam, 1969
• [6] Frobenius F.G., Über lineare Substitutionen und bilineare Formen, J. Reine Angew. Math., 1878, 84, 1–63 http://dx.doi.org/10.1515/crll.1878.84.1
• [7] Gamelin T.W., Uniform Algebras, 2nd ed., Chelsea, New York, 1984
• [8] Hurwitz A., Über die Composition der quadratischen Formen von beliebig vielen Variablen, Nachrichten von der Kgl. Gesellschaft der Wissenschaften zu Göttingen, 1898, 309–316
• [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
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