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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), 639-659
MSC: Primary 46J15; Secondary 46B20, 46G20
MathSciNet review: 986022
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Abstract: Let $ A(B)$ be the uniform algebra on the unit ball of a dual Banach space $ \mathcal{Z} = {\mathcal{Y}^\ast}$ generated by the weak-star continuous linear functionals. We focus on three related problems: (i) to determine when $ A(B)$ is a tight uniform algebra; (ii) to describe which functions in $ {H^\infty }(B)$ are approximable pointwise on $ B$ by bounded nets in $ A(B)$; and (iii) to describe the weak topology of $ B$ regarded as a subset of the dual of $ A(B)$. With respect to the second problem, we show that any polynomial in elements of $ {\mathcal{Y}^{\ast\ast}}$ can be approximated pointwise on $ B$ by functions in $ A(B)$ 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 $ \Lambda $-spaces, with the property that if $ \{ {x_j}\} $ is a bounded sequence in $ \mathcal{X}$ such that $ P({x_j}) \to 0$ for any $ m$-homogeneous analytic function $ P$ on $ \mathcal{X}, m \geq 1$, then $ {x_j} \to 0$ in norm. We show for instance that a Banach space has the Schur property if and only if it is a $ \Lambda $-space with the Dunford-Pettis property.

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  • [1] E. L. Arenson, Gleason parts and the Choquet boundary of a function algebra on a convex compactum, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 204-207. MR 629841 (83d:46059)
  • [2] B. Cole and T.W. Gamelin, Tight uniform algebras, J. Funct. Anal. 46 (1982), 158-220. MR 660185 (83h:46065)
  • [3] -, Weak-star continuous homomorphisms and a decomposition of orthogonal measures, Ann. Inst. Fourier (Grenoble) 35 (1985), 149-189. MR 781784 (86m:46051)
  • [4] D. W. Dean, The equation $ L(E,{X^{\ast\ast}}) = L{(E,X)^{\ast\ast}}$ and the principle of local reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146-148. MR 0324383 (48:2735)
  • [5] J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, Berlin, 1984. MR 737004 (85i:46020)
  • [6] -, A survey of results related to the Dunford-Pettis 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. 15-60. MR 621850 (82i:46023)
  • [7] T. W. Gamelin, Uniform algebras, 2nd ed., Chelsea, New York, 1984.
  • [8] -, Uniform algebras on plane sets, Approximation Theory, G. G. Lorentz (ed.), Academic Press, 1973, pp. 100-149. MR 0338784 (49:3548)
  • [9] J. Lindenstrauss and L. Tsafriri, Classical Banach spaces, Vol. 1, Springer-Verlag, Berlin, 1977.
  • [10] J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. 1 (1930), 164-174.
  • [11] L. Nachbin, Topology on spaces of holomorphic mappings, Springer-Verlag, Berlin, 1969. MR 0254579 (40:7787)
  • [12] A. Pelczynski, A property of multilinear operations, Studia Math. 16 (1957), 173-182. MR 0093698 (20:221)
  • [13] 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), 174-180.

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