Boundaries for algebras of holomorphic functions

Moraes, Luiza A.; Grados, Luis Romero

doi:10.1016/s0022-247x(03)00150-1

Cited by 15 publications

(19 citation statements)

References 4 publications

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“…Aron, Choi, Lourenço and Paques [3] studied the case p (1 p ∞). Later Moraes and Romero Grados [11] considered the case of the space d * (w, 1) for w = {1/n}, the canonical predual of the Lorentz sequence space d(w, 1).…”

Section: Introductionmentioning

confidence: 99%

“…The proofs given in [11] and [12] work for every weight sequence w. Recently Choi and Han [6] considered the case X = M w 0 (a space whose bidual is the Marcinkiewicz sequence space). If w is a decreasing sequence of positive numbers such that w ∈ c 0 \ l 1 , the space M w 0 coincides with the space d * (w, 1) and the results obtained in [6] generalize part of the results obtained by Moraes and Romero Grados in [11]. Choi, García, Kim and Maestre [5] proved that for K scattered and X = C(K), the subset of extreme points of the unit ball of X is a boundary for A u (B X ).…”

Section: Introductionmentioning

confidence: 99%

“…We start by presenting (without proof) a formal generalization of Theorem 3.5 in [11] to the predual of any Lorentz sequence space d(w, 1). This result gives a characterization of the boundaries for A u (B d * (w,1) ) in terms of the finite-dimensional projections.…”

Section: Introductionmentioning

confidence: 99%

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On boundaries on the predual of the Lorentz sequence space

Acosta

Moraes

Grados

2007

Journal of Mathematical Analysis and Applications

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Let X be the canonical predual of the Lorentz sequence space and let A u (B X ) be the Banach algebra of all complex valued functions defined on the closed unit ball B X of X which are uniformly continuous on B X and holomorphic on the interior of B X , endowed with the sup norm. A characterization of the boundaries for A u (B X ) is given in terms of the distance to the strong peak sets of this algebra.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On boundaries on the predual of the Lorentz sequence space

Acosta

Moraes

Grados

2007

Journal of Mathematical Analysis and Applications

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“…Moraes and Romero [14] gave a characterization of the boundaries of A u (B d * (w,1) ), where d * (w, 1) is the canonical predual of the Lorentz sequence space d(w, 1) when w = (1/n). Later Acosta, Moraes and Romero [2] generalized that characterization proving it for any space d * (w, 1) and obtained another one in terms of the strong peak sets of the unit ball.…”

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confidence: 99%

Shilov boundary for holomorphic functions on some classical Banach spaces

Acosta

Lourenço

2007

Studia Math.

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Abstract. Let A ∞ (B X ) be the Banach space of all bounded and continuous functions on the closed unit ball B X of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let A u (B X ) be the subspace of A ∞ (B X ) of those functions which are uniformly continuous on B X . A subset B ⊂ B X is a boundary for A ∞ (B X ) if f = sup x∈B |f (x)| for every f ∈ A ∞ (B X ). We prove that for X = d(w, 1) (the Lorentz sequence space) and X = C 1 (H), the trace class operators, there is a minimal closed boundary for A ∞ (B X ). On the other hand, for X = S, the Schreier space, and, there is no minimal closed boundary for the corresponding spaces of holomorphic functions.

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“…The pioneering work of Globevnik was followed by many authors (see, e.g., [6], [27], [28], [13], [1], [2], [14], [4], [15] and [12]), who studied the existence and characterization of such generalized boundaries. These authors considered algebras of holomorphic mappings defined on the closed unit ball of concrete complex Banach spaces which were, by and large, sequence spaces.…”

Section: Introductionmentioning

confidence: 99%

On boundaries for spaces of holomorphic functions on the unit ball of a Banach space

Acosta¹,

Moraes²

Banach Spaces and Their Applications in Analysis

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Abstract. For a complex Banach space X, let A u (B X ) be the Banach algebra of all complex valued functions defined on B X that are uniformly continuous on B X and holomorphic on the interior of B X , and let A wu (B X ) be the Banach subalgebra consisting of those functions in A u (B X ) that are uniformly weakly continuous on B X . In this paper we study a generalization of the notion of boundary for these algebras, originally introduced by Globevnik. In particular, we characterize the boundaries of A wu (B X ) when the dual of X is separable. We exhibit some natural examples of Banach spaces where this characterization provides concrete criteria for the boundary. We also show that every non-reflexive Banach space X which is an M-ideal in its bidual cannot have a minimal closed boundary for A u (B X ).

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Boundaries for algebras of holomorphic functions

Cited by 15 publications

References 4 publications

On boundaries on the predual of the Lorentz sequence space

On boundaries on the predual of the Lorentz sequence space

Shilov boundary for holomorphic functions on some classical Banach spaces

On boundaries for spaces of holomorphic functions on the unit ball of a Banach space

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