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## Open Mathematics

2010 | 8 | 5 | 878-889
Tytuł artykułu

### Multiplicatively and non-symmetric multiplicatively norm-preserving maps

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EN
Abstrakty
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
878-889
Opis fizyczny
Daty
wydano
2010-10-01
online
2010-09-28
Twórcy
autor
• Tarbiat Modares University
autor
• Tarbiat Modares University
Bibliografia
•  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
•  Browder A., Introduction to Function Algebras, Mathematics Lecture Note Series, W.A. Benjamin, New York-Amsterdam, 1969
•  Burgos M., Jiménez-Vargas A., Villegas-Vallecillos M., Nonlinear conditions for weighted composition operators between Lipschitz algebras, J. Math. Anal. Appl., 2009, 359(1), 1–14 http://dx.doi.org/10.1016/j.jmaa.2009.05.017
•  Dales H.G., Boundaries and peak points for Banach function algebras, Proc. Lond. Math. Soc., 1971, 22, 121–136 http://dx.doi.org/10.1112/plms/s3-22.1.121
•  Hatori O., Miura T., Oka H., Takagi H., Peripheral multiplicativity of maps on uniformly closed algebras of continuous functions which vanish at infinity, Tokyo J. Math, 2009, 32(1), 91–104 http://dx.doi.org/10.3836/tjm/1249648411
•  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
•  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
•  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
•  Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2007, 435, 199–205
•  Honma D., Norm-preserving surjections on algebras of continuous functions, Rocky Mountain J. Math., 2009, 39(5), 1517–1531 http://dx.doi.org/10.1216/RMJ-2009-39-5-1517
•  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
•  Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., (in press)
•  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
•  Lambert S., Spectral Preserver Problems in Uniform Algebras, Ph.D. thesis, University of Montana, Missoula, 2008
•  Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math., 2007, 435, 265–281
•  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
•  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
•  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
•  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
•  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
•  Rao N.V., Tonev T.V., Toneva E.T., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416
•  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
•  Stout E.L., The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson, 1971
•  Tonev T., Weakly multiplicative operators on function algebras without units, Banach Center Publ., (in press)
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