A separable L-embedded Banach space has property (X) and is therefore the unique predual of its dual

We say that a Banach space X is the unique predual of its dual (more precisely the unique isometric predual of its dual) in case it is isometric to any Banach space whose dual is isometric to the dual of X. (We say that two Banach spaces Y and Z are isomorphic if there is a bounded linear bijective operator T : Y → Z with bounded inverse T; if moreover ‖T (y)‖ = ‖y‖ for all y ∈ Y we say that Y and Z are isometric.) In general a Banach space need not be the unique predual of its dual, for example c and c0 are not isometric Banach spaces although their duals are. As shown by Grothendieck [10, Rem. 4] in 1955, L-spaces are unique preduals of their duals. Using essentially a result of Dixmier [5] from 1953, Sakai [19, Cor. 1.13.3] observed that more generally preduals of von Neumann algebras are unique, and Barton and Timoney [2] and Horn [13] generalized this to preduals of JBW triples. Ando [1] stated the uniqueness as a predual for the quotient L/H 0 . As Banach spaces these examples have in common to be L-summands in their biduals or, for short, to be L-embedded. By definition a Banach space X is Lembedded if there is a projection P on its bidual X with range X such that ‖Px‖+ ‖x − Px‖ = ‖x‖ for all x ∈ X. The standard reference for L-embedded spaces is [11], for a survey on unique preduals we refer to [8], for general Banach space theory to [14], [15], or [4]. If not stated otherwise a sequence (zj) (and similarly a series ∑