On \(C(K)\) Grothendieck spaces
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
scientific article; zbMATH DE number 5121515
scientific article