An extension property for Banach spaces

Walden Freedman¹

¹ Department of Mathematics College of Arts and Sciences Koç University Rumelifeneri Yolu Sariyer, Istanbul, Turkey Current address Department of Mathematics Humboldt State University 1 Harpst Street Arcata, CA 95521, U.S.A.

Colloquium Mathematicum, Tome 91 (2002) no. 2, pp. 167-182

Résumé

A Banach space $X$ has property $(E)$ if every operator from $X$ into $c_0$ extends to an operator from $X^{\ast \ast }$ into $c_0$; $X$ has property $(L)$ if whenever $K\subseteq X$ is limited in $X^{\ast \ast }$, then $K$ is limited in $X$; $X$ has property $(G)$ if whenever $K\subseteq X$ is Grothendieck in $X^{\ast \ast }$, then $K$ is Grothendieck in $X$. In all of these, we consider $X$ as canonically embedded in $X^{\ast \ast }$. We study these properties in connection with other geometric properties, such as the Phillips properties, the Gelfand–Phillips and weak Gelfand–Phillips properties, and the property of being a Grothendieck space.

Détail

DOI : 10.4064/cm91-2-2

Keywords: banach space has property every operator extends operator ast ast has property whenever subseteq limited ast ast limited has property whenever subseteq grothendieck ast ast grothendieck these consider canonically embedded ast ast study these properties connection other geometric properties phillips properties gelfand phillips weak gelfand phillips properties property being grothendieck space