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We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from 1 into Y . Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L 1 (μ)-space for a σ -finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.

Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series a n /n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients a n in Y is bounded by 1 − 1/ Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y . This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. Mathematics Subject Classification (2000)Primary 32A05; Secondary 46B07 · 46B09 · 46G20 The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210. A. Defant

Abstract. We investigate uniform algebras of bounded analytic functions on the unit ball of a complex Banach space. We prove several cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These lead to weak versions of the corona theorem for the open unit balls of c 0 and 2 , in which all but one of the functions comprising the corona data extend to be weak-star continuous on the closed unit ball.

Each bounded holomorphic function on the infinite dimensional polydisk D ∞

Let U and V be convex and balanced open subsets of the Banach spaces X and Y respectively. In this paper we study the following question: Given two Fréchet algebras of holomorphic functions of bounded type on U and V respectively that are algebra-isomorphic, can we deduce that X and Y (or X * and Y *) are isomorphic? We prove that if X * or Y * has the approximation property and H wu (U) and H wu (V) are topologically algebra-isomorphic, then X * and Y * are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H b (U) and H b (V), giving conditions under which an algebra-isomorphism between H b (X) and H b (Y) is equivalent to an isomorphism between X * and Y *. We also obtain characterizations of different algebra-homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H b (X) with pathological behaviors.

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Abstract. A Banach space E is known to be Arens regular if every continuous linear mapping from E to E is weakly compact. Let U be an open subset of E, and let H b (U ) denote the algebra of analytic functions on U which are bounded on bounded subsets of U lying at a positive distance from the boundary of U. We endow H b (U ) with the usual Fréchet topology. M b (U ) denotes the set of continuous homomorphisms φ : H b (U ) → C. We study the relation between the Arens regularity of the space E and the structure of M b (U ).