\(\lambda\) property for Bochner--Orlicz sequence spaces with Orlicz norm (Q2518176)

The authors give a complete characterization of the \(\lambda\) property and the uniform \(\lambda\) property for Bochner--Orlicz sequence spaces equipped with Orlicz norm \(l_M (X)\). If \(X\) is a Banach space, \(B(X)\) is its unit ball and \(\text{Ext}B(X)\) the set of the extreme points of \(B(X)\), for an \(x \in B(X)\), \(\lambda(x)\) is defined by \[ \lambda(x)= \sup\{\lambda\in [0,1]:x=\lambda e+(1-\lambda)y,\;y \in B(X),\;e \in\text{Ext}B(X)\}. \] If \(\lambda(x) >0\), it is said that \(x\) is a \(\lambda\) point. \(X\) has the \(\lambda\) property if \(\lambda(x)>0\) for all \(x \in B(X)\), and the uniform \(\lambda\) property if \(\lambda(X) >0\) where \(\lambda(X)= \inf \{\lambda(x):x\in B(X)\}\). The main results of the paper establish that \(l_M(X)\) has the \(\lambda\) property if and only if the Banach space \(X\) has the uniform \(\lambda\) property (Theorem 1), and the uniform \(\lambda\) property if and only if \(X\) has it too and another uniformity condition on the function \(M\) is satisfied (Theorem 2). As a consequence of these results, it is obtained that the \(\lambda\) property cannot be lifted from \(X\) to \(l_M(X)\).