An extension property for Banach spaces. (English) Zbl 1028.46020

Freedman, Walden

Colloq. Math. 91, No. 2, 167-182 (2002).

The authors considers an extension property called ‘property \((E)\)’. A Banach space \(X\) has this property if every bounded linear operator from \(X\) into \(c_0\) extends to a bounded linear operator from \(X^{**}\) into \(c_0\). In Theorem 2.2, the author gives several equivalent formulations, one of which is that weak\(^*\)-null sequences in \(X^*\) lift to weak\(^*\)-null sequences in \(X^{***}\). In Section 3 of the paper the author among other things considers the weak Phillips property. A Banach space \(X\) has this property if for every operator \(T:X^{**}\to c_0\), its restriction to \(X\) is weakly compact. Theorem 3.6 which gives various equivalent formulations of Grothendieck spaces (i.e., spaces where weak\(^*\) and weak sequential convergences coincide in the dual) shows that \(X\) is a Grothendieck space if and only if it has the properties \((E)\) and weak Phillips.

Reviewer: T.S.S.R.K.Rao (Bangalore)

MSC:

46B03	Isomorphic theory (including renorming) of Banach spaces
46B20	Geometry and structure of normed linear spaces
46B25	Classical Banach spaces in the general theory

Keywords:

weak Phillips property; property (V*); Grothendieck spaces; Sobczyk’s Theorem; extension property; lifting property