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2013 | 11 | 7 | 1197-1211
Tytuł artykułu

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

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EN
Abstrakty
EN
Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb{K}$-valued Lipschitz functions on X - where $\mathbb{K}$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f‖∞} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb{K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.
Twórcy
  • Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Almería, 04120, Spain, ajimenez@ual.es
  • Department of Mathematics, Iowa State University, Ames, IA, 50011, USA, leekm@iastate.edu
  • Mathematics and Software Development, National Security Technologies, LLC, P.O. Box 98521, M/S NLV078, Las Vegas, NV, 89193, USA, aluttman21@gmail.com
Bibliografia
  • [1] Browder A., Introduction to Function Algebras, W.A. Benjamin, New York-Amsterdam, 1969
  • [2] Hatori O., Lambert S., Luttman A., Miura T., Tonev T., Yates R., Spectral preservers in commutative Banach algebras, Edwardsville, May 18–22, 2010, In: Function Spaces in Modern Analysis, Contemp. Math., 547, American Mathematical Socitey, Providence, 2011, 103–123 http://dx.doi.org/10.1090/conm/547/10812[Crossref]
  • [3] Hatori O., Miura T., Shindo R., Takagi T., Generalizations of spectrally multiplicative surjections between uniform algebras, Rend. Circ. Mat. Palermo, 2010, 59(2), 161–183 http://dx.doi.org/10.1007/s12215-010-0013-3[Crossref]
  • [4] Hatori O., Miura T., Takagi H., Characterization of isometric isomorphisms between uniform algebras via non-linear range preserving properties, Proc. Amer. Math. Soc., 2006, 134, 2923–2930 http://dx.doi.org/10.1090/S0002-9939-06-08500-5[Crossref]
  • [5] Hatori O., Miura T., Takagi H., Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl., 2007, 326(1), 281–296 http://dx.doi.org/10.1016/j.jmaa.2006.02.084[WoS][Crossref]
  • [6] Hatori O., Miura T., Takagi H., Polynomially spectrum-preserving maps between commutative Banach algebras, preprint available at http://arxiv.org/abs/0904.2322
  • [7] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, In: Function Spaces, Edwardsville, May 16–20, 2006, Contemp. Math., 435, American Mathematical Society, Providence, 2007, 199–205
  • [8] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., 2010, 40(3), 1903–1922 http://dx.doi.org/10.1216/RMJ-2010-40-6-1903[Crossref][WoS]
  • [9] Jiménez-Vargas A., Villegas-Vallecillos M., Lipschitz algebras and peripherally-multiplicative maps, Acta Math. Sin. (Engl. Ser.), 2008, 24(8), 1233–1242 http://dx.doi.org/10.1007/s10114-008-7202-4[Crossref]
  • [10] Lee K., Luttman A., Generalizations of weakly peripherally multiplicative maps between uniform algebras, J. Math. Anal. Appl., 2011, 375(1), 108–117 http://dx.doi.org/10.1016/j.jmaa.2010.08.051[Crossref]
  • [11] Luttman A., Lambert S., Norm conditions and uniform algebra isomorphisms, Cent. Eur. J. Math., 2008, 6(2), 272–280 http://dx.doi.org/10.2478/s11533-008-0016-x[WoS][Crossref]
  • [12] Luttman A., Tonev T., Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 2007, 135(11), 3589–3598 http://dx.doi.org/10.1090/S0002-9939-07-08881-8[WoS][Crossref]
  • [13] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2001, 130(1), 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X[Crossref]
  • [14] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 2005, 133(4), 1135–1142 http://dx.doi.org/10.1090/S0002-9939-04-07615-4[Crossref]
  • [15] Shindo R., Weakly-peripherally multiplicative conditions and isomorphisms between uniform algebras, Publ. Math. Debrecen, 2011, 78(3-4), 675–685 http://dx.doi.org/10.5486/PMD.2011.4937[Crossref][WoS]
  • [16] Tonev T., Yates R., Norm-linear and norm-additive operators between uniform algebras, J. Math. Anal. Appl., 2009, 357(1), 45–53 http://dx.doi.org/10.1016/j.jmaa.2009.03.039[Crossref]
  • [17] Weaver N., Lipschitz Algebras, World Scientific, River Edge, 1999 http://dx.doi.org/10.1142/4100[Crossref]
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0243-7
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