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A uniform algebra of analytic functions on a Banach space. (English) Zbl 0704.46033

Let \({\mathcal Y}\) be a complex Banach space, \({\mathcal Z}={\mathcal Y}^*\) its dual, B the open unit ball of \({\mathcal Z}\). This interesting and suggestive paper studies A(B), the uniform algebra on \(\bar B\) generated by weak- star continuous linear functionals on \({\mathcal Z}\), that is, by the elements of \({\mathcal Y}\) regarded as functions on B; if \({\mathcal Y}={\mathcal Z}={\mathbb{C}}\) then B is the usual open unit disc and A(B) is the usual disc algebra. The primary objective is to try to understand when A(B) is “tight”, a notion introduced by B.J. Cole and T. W. Gamelin in an earlier paper [J. Funct. Anal. 46, 158-220 (1982; Zbl 0569.46034)]. A uniform algebra A on a compact Hausdorff space X is said to be “tight” if for each \(g\in C(X)\) the Hankel-type operator \(S_ g: A\to C(X)/A\) given by \(S_ g(f)=gf+A\) is weakly compact. A sample result about tightness is Theorem 9.1: If A(B) is tight then \({\mathcal Z}\) is reflexive.
Most of the paper is occupied by preliminary results of interest in their own right. In particular, it turns out that it is important to understand the \(A(B)^{**}\)-topology of B \((B\subset A(B)^*\) canonically). It develops that \(A(B)^{**}|_ B=A_{pb}(B)\), the algebra of functions on B which are approximable pointwise by bounded nets in A(B). Always \(A_{pb}(B)\subset H^{\infty}(B)\), the algebra of bounded analytic functions on B; it turns out (Theorem 4.4) that this inclusion is equality if \({\mathcal Z}\) has the metric approximation property \((T_{\alpha}x\to x\) for all \(x\in {\mathcal Z}\), for some net \(\{T_{\alpha}\}\) of finite rank operators \(T_{\alpha}\) such that \(\| T_{\alpha}\| \leq 1).\)
Consideration of the “weak polynomial topology” leads to the notion of “\(\Lambda\)-space”, one in which if a sequence \(\{x_ j\}\) satisfies \(f(x_ j)\to 0\) for all analytic polynomials f, then \(\| x_ j\| \to 0\); any Banach space which enjoys the standard Schur property is a \(\Lambda\)-space. Results are given about when a classical Banach space is or is not a \(\Lambda\)-space (sample undecided case: \(L^ p[0,1]\), \(1<p<2)\), and Theorem 7.5 says that having the Schur property is equivalent to being a \(\Lambda\)-space and having the Dunford-Pettis property.
Reviewer: S.J.Sidney

MSC:

46J10 Banach algebras of continuous functions, function algebras
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46B10 Duality and reflexivity in normed linear and Banach spaces

Citations:

Zbl 0569.46034
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References:

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