On \(C(K)\) Grothendieck spaces
type: Item
instance of: scholarly article
title: On \(C(K)\) Grothendieck spaces
zbMATH Open document ID: 1116.46014
DOI: 10.1007/BF02874635
author: Diomedes Barcenas
author: Luis Gerardo Mármol
publication date: 2007-02-02T00:00:00Z
review text: The main result in this paper is a list of ten equivalent formulations of the statement `` \(C(K)\) is a Grothendieck space.''The paper is inspired by \textit{H. P. Lotz} [Math. Z. 190, 207--220 (1985; Zbl 0623.47033)], where many properties of Grothendieck spaces having also the Dunford--Pettis property are uncovered, in the sense that several of the results from that paper are shown to have a converse in the \(C(K)\) case. Combining one of these theorems with a result of \textit{J. M. A. M. Van Neerven} [Proc. Am. Math. Soc. 116, No. 2, 525--527 (1992; Zbl 0895.47031)] one obtains one equivalent statement: ``Every strongly continuous semigroup on \(C(K)\) has bounded generator.'' Using this the authors deduce e.g. that `` No infinite-dimensional WCG space is complemented in \(C(K)\)''\ is also an equivalent statement. On the other hand, already ``\(c_o\) is not complemented in \(C(K)\)'' assures that \(C(K)\) is a Grothendieck space. The paper ends with a section containing some examples and remarks
reviewed by: Olav Nygaard
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5121515
zbMATH Keywords: Grothendieck spaces
zbMATH Keywords: Dunford-Pettis property
MaRDI profile type: Publication
cites work: On the Dunford-Pettis property of the tensor product of \(C(K)\) spaces
cites work: New Banach space properties of the disc algebra and \(H^{\infty}\)
cites work: On the Dunford-Pettis Property
cites work: C(K, E) Contains a Complemented Copy of c 0
cites work: Grothendieck's property in Lp(μ, X)
cites work: Q3688898
cites work: Q4076052
cites work: Q3329946
cites work: Q4431835
cites work: Q4148368
cites work: Grothendieck Spaces and Duals of Injective Tensor Products
cites work: Bases of tensor products of Banach spaces
cites work: Sur Les Applications Lineaires Faiblement Compactes D'Espaces Du Type C(K)
cites work: Q4373847
cites work: No Infinite Dimensional P Space Admits a Markuschevich Basis
cites work: Uniform convergence of operators and Grothendieck spaces with the Dunford-Pettis property
cites work: Uniform convergence of operators on \(L^{\infty}\) and similar spaces
cites work: Banach lattices
cites work: A Converse of Lotz's Theorem on Uniformly Continuous Semigroups
cites work: Q4049667
full work available at URL: https://doi.org/10.1007/bf02874635
OpenAlex ID: W2030637055
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5121515