Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Maliheh Hosseini ; Fereshteh Sady

Open Mathematics, Tome 8 (2010), p. 878-889 / Harvested from The Polish Digital Mathematics Library

Résumé

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, Tfy=ηyfφyy∈K,-ααηyfφy¯y∈cB∖K . 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.

Détail

Publié le : 2010-01-01

Zbl 1229.46034

EUDML-ID : urn:eudml:doc:269046

Bibliographie

[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 | Zbl 0870.54018

[2] Browder A., Introduction to Function Algebras, Mathematics Lecture Note Series, W.A. Benjamin, New York-Amsterdam, 1969 | Zbl 0199.46103

[3] 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 | Zbl 1181.47037

[4] 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 | Zbl 0211.15902

[5] 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 | Zbl 1201.46046

[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 | Zbl 1102.46032

[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 | Zbl 1113.46047

[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

[9] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2007, 435, 199–205 | Zbl 1141.46324

[10] 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 | Zbl 1183.46051

[11] 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 | Zbl 1171.46021

[12] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., (in press) | Zbl 1220.46033

[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 | Zbl 1178.46049

[14] Lambert S., Spectral Preserver Problems in Uniform Algebras, Ph.D. thesis, University of Montana, Missoula, 2008

[15] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math., 2007, 435, 265–281 | Zbl 1148.46030

[16] 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 | Zbl 1151.46036

[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 | Zbl 1134.46030

[18] 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 | Zbl 0983.47024

[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 | Zbl 1068.46028

[20] 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 | Zbl 1074.46033

[21] Rao N.V., Tonev T.V., Toneva E.T., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416 | Zbl 1123.46035

[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 | Zbl 1201.47039

[23] Stout E.L., The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson, 1971 | Zbl 0286.46049

[24] Tonev T., Weakly multiplicative operators on function algebras without units, Banach Center Publ., (in press)