Characterizations of complex \(L^ 1\)-preduals. (English) Zbl 0568.46015
A complex Banach space A is called an E(n)-space if, given \(\{B(a_ k,r_ k)\}^ n_{k=1}\) of n closed balls in A such that \(\cap^{n}_{k=1}B(f(a_ k),r_ k)\neq \emptyset\) for all \(f\in A^*\), \(\| f\| \leq 1\), then \(\cap^{n}_{k=1}B(a_ k,r_ k)\neq \emptyset\) in A. We give a self-contained proof that complex E(3)-spaces are \(L^ 1\)-preduals. The main results can be summarized as follows.
Assume F is a proper face of the dual unit ball of A. Then we have: 1) F is a split face of conv(FU-iF). 2) \(cone F\) has the Riesz decomposition property. 3) \(lin F\) is isometrically isomorphic to a complex \(L^ 1(\mu)\)-space. 4) \(lin F\) is an L-summand.
Assume F is a proper face of the dual unit ball of A. Then we have: 1) F is a split face of conv(FU-iF). 2) \(cone F\) has the Riesz decomposition property. 3) \(lin F\) is isometrically isomorphic to a complex \(L^ 1(\mu)\)-space. 4) \(lin F\) is an L-summand.
MSC:
| 46B25 | Classical Banach spaces in the general theory |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B10 | Duality and reflexivity in normed linear and Banach spaces |
| 46B42 | Banach lattices |
