Maps between Banach function algebras satisfying certain norm conditions

Maliheh Hosseini ; Fereshteh Sady

Open Mathematics, Tome 11 (2013), p. 1020-1033 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1275.46035

EUDML-ID : urn:eudml:doc:269472

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[Crossref] | Zbl 0870.54018

[2] Dales H.G., Boundaries and peak points for Banach function algebras, Proc. London Math. Soc., 1971, 22(1), 121–136 http://dx.doi.org/10.1112/plms/s3-22.1.121[Crossref] | Zbl 0211.15902

[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] | Zbl 1239.46036

[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] | Zbl 1209.46027

[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] | Zbl 1102.46032

[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] | Zbl 1113.46047

[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] | Zbl 1171.46021

[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] | Zbl 1229.46034

[10] Hosseini M., Sady F., Banach function algebras and certain polynomially norm-preserving maps, Banach J. Math. Anal., 2012, 6(2), 1–18 | Zbl 1256.46030

[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] | Zbl 1220.46033

[12] Kowalski S., Słodkowski Z., A characterization of multiplicative linear functionals in Banach algebras, Studia Math., 1980, 67(3), 215–223 | Zbl 0456.46041

[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] | Zbl 1269.46033

[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 | Zbl 1148.46030

[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] | Zbl 1151.46036

[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] | Zbl 1134.46030

[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] | Zbl 0983.47024

[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] | Zbl 1068.46028

[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] | Zbl 1074.46033

[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] | Zbl 1222.46039

[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] | Zbl 1201.47039

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

[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 | Zbl 1217.46034