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2013 | 11 | 6 | 1020-1033
Tytuł artykułu

Maps between Banach function algebras satisfying certain norm conditions

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Abstrakty
EN
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let \(\bar A\) and \(\bar B\) be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A, [...] where \(\hat \cdot\) denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from \(\bar A\) onto \(\bar B\) inducing a homeomorphism between \(M_{\bar B}\) and \(M_{\bar A}\) . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
6
Strony
1020-1033
Opis fizyczny
Daty
wydano
2013-06-01
online
2013-03-28
Twórcy
  • Department of Mathematics, K.N. Toosi University of Technology, 16315-1618, Tehran, Iran, m.hosseini@kntu.ac.ir
  • Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, 14115-134, Tehran, Iran, sady@modares.ac.ir
Bibliografia
  • [1] Araujo J., Font J.J., On Šilov boundaries for subspaces of continuous functions, Topology Appl., 1997, 77(2), 79–85 http://dx.doi.org/10.1016/S0166-8641(96)00132-0[Crossref]
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  • [3] Hatori O., Lambert S., Luttman A., Miura T., Tonev T., Yates R., Spectral preservers in commutative Banach algebras, In: Function Spaces in Modern Analysis, Edwardsville, May 18–22, 2010, Contemp. Math., 547, American Mathemtical Society, Providence, 2011, 103–124 http://dx.doi.org/10.1090/conm/547/10812[Crossref]
  • [4] Hatori O., Miura T., Shindo R., Takagi H., 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]
  • [5] Hatori O., Miura T., Takagi H., Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties, Proc. Amer. Math. Soc., 2006, 134(10), 2923–2930 http://dx.doi.org/10.1090/S0002-9939-06-08500-5[Crossref]
  • [6] 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]
  • [7] Hatori O., Miura T., Takagi H., Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras, preprint available at http://arxiv.org/abs/0904.1939
  • [8] Hosseini M., Sady F., Multiplicatively range-preserving maps between Banach function algebras, J. Math. Anal. Appl., 2009, 357(1), 314–322 http://dx.doi.org/10.1016/j.jmaa.2009.04.008[WoS][Crossref]
  • [9] Hosseini M., Sady F., Multiplicatively and non-symmetric multiplicatively norm-preserving maps, Cent. Eur. J. Math., 2010, 8(5), 878–889 http://dx.doi.org/10.2478/s11533-010-0053-0[Crossref][WoS]
  • [10] Hosseini M., Sady F., Banach function algebras and certain polynomially norm-preserving maps, Banach J. Math. Anal., 2012, 6(2), 1–18
  • [11] 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[Crossref][WoS]
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  • [13] Lambert S., Luttman A., Generalized strong boundary points and boundaries of families of continuous functions, Mediterr. J. Math., 2012, 9(2), 337–355 http://dx.doi.org/10.1007/s00009-010-0105-5[Crossref][WoS]
  • [14] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, In: Function Spaces, Edwardsville, May 16–20, 2006, Contemp. Math., 435, American Mathematical Society, Providence, 2007, 265–281
  • [15] Luttman A., Lambert S., Norm conditions for uniform algebra isomorphisms, Cent. Eur. J. Math., 2008, 6(2), 272–280 http://dx.doi.org/10.2478/s11533-008-0016-x[WoS][Crossref]
  • [16] 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]
  • [17] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2002, 130(1), 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X[Crossref]
  • [18] 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]
  • [19] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc., 2005, 48(1), 219–229 http://dx.doi.org/10.1017/S0013091504000719[Crossref]
  • [20] Shindo R., Maps between uniform algebras preserving norms of rational functions, Mediterr. J. Math., 2010, 8(1), 81–95 http://dx.doi.org/10.1007/s00009-010-0056-x[WoS][Crossref]
  • [21] Shindo R., Norm conditions for real-algebra isomorphisms between uniform algebras, Cent. Eur. J. Math., 2010, 8(1), 135–147 http://dx.doi.org/10.2478/s11533-009-0060-1[Crossref][WoS]
  • [22] Stout E.L., The Theory of Uniform Algebras, Bogden & Quigley, Tarrytown-on-Hudson, 1971
  • [23] Tonev T., Weak multiplicative operators on function algebras without units, In: Banach Algebras, Bedlewo, July 14–24, 2009, Banach Center Publ., 91, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 411–421
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0224-x
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