Abstract
Let K be a Hausdorff space and C b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of C b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C b (K) is a dense G δ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.
In the case that X is a complex Banach space with the Radon-Nikodým property, we show that the set of all strong peak functions in A b (B X ) = {f ∈ C b (B X ): f|B ∘ X is holomorphic} is dense. As an application, we show that the smallest closed norming subset of A b (B X ) is the closure of the set of all strong peak points for A b (B X ). This implies that the norm of A b (B X ) is Gâteaux differentiable on a dense subset of A b (B X ), even though the norm is nowhere Fréchet differentiable when X is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of the numerical Shilov boundary.
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M. D. Acosta, J. Alaminos, D. García and M. Maestre, On holomorohic functions attaining their norms, Journal of Mathematical Analysis and Applications 297 (2004), 625–644.
M. D. Acosta and S. G. Kim, Numerical boundaries for some classical Banach spaces, Journal of Mathematics and Applications 350 (2009), 694–707.
R. M. Aron, Y. S. Choi, M. L. Lourenço and Q. W. Paques, Boundaries for algebras of analytic functions on infinite-dimensional Banach spaces. Banach spaces (Merida, 1992), Contemporary Mathematics 144, American Mathematical Society, Providence, RI, 1993, pp. 15–22.
R. M. Aron, B. J. Cole and T. W. Gamelin, Spectra of algebras of analytic functions on a Banach space, Journal für die Reine und Angewandte Mathematik 415 (1991), 51–93.
E. Bishop, A minimal boundary for function algebras, Pacific Journal of Mathematics 9 (1959), 629–642.
J. Bourgain, On dentability and the Bishop-Phelps property, Israel Journal of Mathematics 28 (1977), 265–271.
A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in a Banach space, Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki 31 (1982), no. 2, 203–214, 317; English translation: Mathematical Notes 31 (1982), 104–110.
T. K. Carne, B. Cole and T. W. Gamelin, A uniform algebra of analytic functions on a Banach space, Transactions of the American Mathematical Society 314 (1989), 639–659.
S. B. Chae, Holomorphy and calculus in normed spaces, Marcel Dekker, Inc., New York, 1985.
Y. S. Choi and K. H. Han, Boundaries for algebras of holomorphic functions on Marcinkiewicz sequence spaces, Journal of Mathematical Analysis and Applications 323 (2006), 1116–1133.
Y. S. Choi, K. H. Han and H. J. Lee, Boundaries for algebras of holomorphic functions on Banach spaces, Illinois Journal of Mathematics 51 (2007), 883–896.
Y. S. Choi and S. G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, Journal of the London Mathematical Society 54 (1996), 135–147.
W. J. Davis, D. J. H. Garling and N. Tomczak-Jagermann, The complex convexity of quasi-normed linear spaces, Journal of Functional Analysis 55 (1984), 110–150.
R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach spaces, John Wiley & Sons, Inc., New York, 1993.
S. Dineen, Complex analysis on infinite-dimensional spaces, Springer-Verlag, London, 1999.
J. Diestel and J. J. Uhl, Vector meausres, American Mathematical Society, Providence, RI, 1977.
P. N. Dowling, Z. Hu and D. Mupasiri, Complex convexity in Lebesgue-Bochner function spaces, Transactions of the American Mathematical Society 348 (1996), 127–139.
J. Ferrera, Norm-attaining polynomials and differentiability, Studia Mathematica 151 (2002), 1–21.
V. P. Fonf, J. Lindenstrauss and R. R. Phelps, Infinite dimensional convexity, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 599–670.
P. Foralewski and P. Kolwicz, Local uniform rotundity in Calderón-Lozanovskiĭ spaces, Journal of Convex Analysis 14 (2007), 395–412.
T. W. Gamelin, Uniform algebras, Chelsea Publishing Company, New York, NY, 1984.
N. Ghoussoub, J. Lindenstrauss and B. Maurey, Analytic martingales and plurisubharmonic barriers in complex Banach spaces, Banach space theory (Iowa City, IA, 1987), 111–130, Contemporary Mathematics 85, American Mathematical Society, Providence, RI, 1989, pp. 111–130.
J. Globevnik, Boundaries for polydisc algebras in infinite dimensions, Mathematical Proceedings of the Cambridge Philosophical Society 85 (1979), 291–303.
J. Globevnik, On complex strict and uniform convexity, Proceedings of the American Mathematical Society 47 (1975), 175–178.
J. Globevnik, On interpolation by analytic maps in infinite dimensions, Mathematical Proceedings of the Cambridge Philosophical Society 83 (1978), 243–252.
L. Romero Grados and L. A. Moraes, Boundaries for algebras of holomorphic functions, Journal of Mathematical Analysis and Applications 281 (2003), 575–586.
H. Hudzik, A. Kamińska and M. Mastyło, On geometric properties of Orlicz-Lorentz spaces, Canadian Mathematical Bulletin 40 (1997), 316–329.
W. B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W. B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84.
R. Larsen, Banach Algebras, Marcel-Dekker, Inc., New York, 1973.
H. J. Lee, Complex convexity and monotonicity in quasi-Banach lattices, Israel Journal of Mathematics 159 (2007), 57–91.
H. J. Lee, Monotonicity and complex convexity in Banach lattices, Journal of Mathematical Analysis and Applications 307 (2005), 86–101.
H. J. Lee, Randomized series and geometry of Banach spaces, Taiwanese Journal of Mathematics, to appear.
A. Rodríguez-Palacios, Numerical ranges of uniformly continuous functions on the unit sphere of a Banach space, Special issue dedicated to John Horváth. Journal of Mathematical Analysis and Applications 297 (2004), 472–476.
Šilov, Sur la théorie des idéaux dans les anneaux normés de fonctions, C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 900–903.
C. Stegall, Optimization and differentiation in Banach spaces, Linear Algebra and its Applications 84 (1986), 191–211.
R. V. Shvydkoy, Geometric aspects of the Daugavet property, Journal of Functional Analysis 176 (2000), 198–212.
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The first named author was supported by grant No. R01-2004-000-10055-0 from the Basic Research Program of the Korea Science & Engineering Foundation and the second named author was supported by the Dongguk University Research Fund of 2008.
The second named author is the corresponding author.
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Choi, Y.S., Lee, H.J. & Song, H.G. Bishop’s theorem and differentiability of a subspace of C b (K). Isr. J. Math. 180, 93–118 (2010). https://doi.org/10.1007/s11856-010-0095-9
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DOI: https://doi.org/10.1007/s11856-010-0095-9