The weak Phillips property (Q2717687)

A Banach space \(X\) is said to have the ``weak Phillips property'' if the natural projection \(X^***\) onto \(X^*\) is weak\(^*\)-weak sequentially continuous. In this paper several characterizes of Banach spaces having this property are proved, and the relationship of the weak Phillips property to other well-known properties of certain Banach spaces is also considered. In particular, the relationship of this property to the more restrictive Grothendieck property is studied in detail. The paper serves, as well, as a good survey of known results concerning the weak Phillips property most of which are collected and described in the introductory section of the paper. The final sections contains a number of interesting open problems in the area.