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Bishop’s theorem and differentiability of a subspace of C b (K)

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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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Authors and Affiliations

  1. Department of Mathematics, POSTECH, Gyeongnam, Republic of Korea

    Yun Sung Choi

  2. Department of Mathemaics Education, Dongguk University-Seoul, Seoul, Republic of Korea

    Han Ju Lee

  3. National Institute for Mathematical Sciences(NIMS), Seoul, Republic of Korea

    Hyun Gwi Song

Authors
  1. Yun Sung Choi
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  2. Han Ju Lee
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  3. Hyun Gwi Song
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Correspondence to Yun Sung Choi.

Additional information

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

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