The universality of ℓ 1 as a dual space

Abstract

Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a L ∞ space Z whose dual is isomorphic to 1 . If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z , so that Z and U are totally incomparable. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.

Quotients of Banach spaces and surjectively universal spaces

“…We notice that there exists a large number of related results found in the literature; see, for instance, [9,14,15,19,23,24,32]. The novelty in Theorem 3 is that, beside functional analytic tools, its proof is enriched with descriptive set theory and the combinatorial machinery developed in [3] and [4].…”

A new isomorphic ℓ1 predual not isomorphic to a complemented subspace of a C (K ) space

“…More precisely we use a dual version of the Bourgain-Delbaen method [10]. The latter has been instrumental to some recent developments in Banach space theory, namely the solution of the scalar-plus-compact problem [5] and the universality of the class of ℓ 1 preduals among spaces with separable dual [11]. Theorem 2.1 gives sufficient conditions on a norming subset of the dual ball of a Banach space with a basis, in order for the space to be L ∞ .…”

Shift invariant preduals of ℓ 1(ℤ)

“…This construction was an inspiration for the recent solution to the scalar-compact problem [2]: this exotic Banach space is also an 1 predual. Indeed, in [15], it is shown that if X is any Banach space with separable dual, then there is an 1 predual E which contains an isomorphic copy of X. In this paper, we do not assume that a predual E of 1 (Z) is isometric, and instead we allow any isomorphism between E * and 1 (Z).…”