A uniform algebra of analytic functions on a Banach space
Authors:
T. K. Carne, B. Cole and T. W. Gamelin
Journal:
Trans. Amer. Math. Soc. 314 (1989), 639659
MSC:
Primary 46J15; Secondary 46B20, 46G20
MathSciNet review:
986022
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Abstract: Let be the uniform algebra on the unit ball of a dual Banach space generated by the weakstar continuous linear functionals. We focus on three related problems: (i) to determine when is a tight uniform algebra; (ii) to describe which functions in are approximable pointwise on by bounded nets in ; and (iii) to describe the weak topology of regarded as a subset of the dual of . With respect to the second problem, we show that any polynomial in elements of can be approximated pointwise on by functions in of the same norm. This can be viewed as a generalization of Goldstine's theorem. In connection with the third problem, we introduce a class of Banach spaces, called spaces, with the property that if is a bounded sequence in such that for any homogeneous analytic function on , then in norm. We show for instance that a Banach space has the Schur property if and only if it is a space with the DunfordPettis property.
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 B. Cole and T.W. Gamelin, Tight uniform algebras, J. Funct. Anal. 46 (1982), 158220. MR 660185 (83h:46065)
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 , Weakstar continuous homomorphisms and a decomposition of orthogonal measures, Ann. Inst. Fourier (Grenoble) 35 (1985), 149189. MR 781784 (86m:46051)
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 D. W. Dean, The equation and the principle of local reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146148. MR 0324383 (48:2735)
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 J. Diestel, Sequences and series in Banach spaces, SpringerVerlag, Berlin, 1984. MR 737004 (85i:46020)
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 , A survey of results related to the DunfordPettis property, Proc. Conference on Integration, Topology and Geometry in Linear Spaces, W. Graves (ed.), Contemp. Math., Vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 1560. MR 621850 (82i:46023)
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 , Uniform algebras on plane sets, Approximation Theory, G. G. Lorentz (ed.), Academic Press, 1973, pp. 100149. MR 0338784 (49:3548)
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 J. Lindenstrauss and L. Tsafriri, Classical Banach spaces, Vol. 1, SpringerVerlag, Berlin, 1977.
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 J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. 1 (1930), 164174.
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 L. Nachbin, Topology on spaces of holomorphic mappings, SpringerVerlag, Berlin, 1969. MR 0254579 (40:7787)
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 A. Pelczynski, A property of multilinear operations, Studia Math. 16 (1957), 173182. MR 0093698 (20:221)
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 G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conf. Ser. in Math., No. 60, Amer. Math. Soc., Providence, R.I., 1986. MR 829919 (88a:47020)
 [14]
 H. R. Pitt, A note on bilinear forms, J. London Math. Soc. 11 (1936), 174180.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909860220
PII:
S 00029947(1989)09860220
Article copyright:
© Copyright 1989 American Mathematical Society
