Polyhedral preduals of ℓ1 and their representing matrices

We present characterizations of polyhedrality (I), (II) and (III) in a separable L1-predual space in terms of its representing matrix. We also continue our study of diagrammatic representations of separable L1-predual spaces and show that in a polyhedral (IV) predual of L1, every M-ideal corresponds to a directed sub-diagram.