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2010 | 8 | 5 | 878-889
Tytuł artykułu

Multiplicatively and non-symmetric multiplicatively norm-preserving maps

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EN
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A, $$ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right. $$. In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.
Wydawca
Czasopismo
Rocznik
Tom
8
Numer
5
Strony
878-889
Opis fizyczny
Daty
wydano
2010-10-01
online
2010-09-28
Twórcy
Bibliografia
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  • [6] 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
  • [7] Hatori O., Miura T., Takagi H., Unital and multiplicatively spectrum-preserving surjections between 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
  • [8] 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
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  • [12] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., (in press)
  • [13] 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
  • [14] Lambert S., Spectral Preserver Problems in Uniform Algebras, Ph.D. thesis, University of Montana, Missoula, 2008
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  • [17] 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
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  • [19] 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
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  • [22] 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
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  • [24] Tonev T., Weakly multiplicative operators on function algebras without units, Banach Center Publ., (in press)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-010-0053-0
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