Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem

Abstract

Abstract. Let SB be the standard coding for separable Banach spaces as subspaces of C(∆). In these notes, we show that if B ⊂ SB is a Borel subset of spaces with separable dual, then the assignment X → X * can be realized by a Borel function B → SB. Moreover, this assignment can be done in such a way that the functional evaluation is still well defined (Theorem 1). Also, we prove a Borel parametrized version of Zippin's theorem, i.e., we prove that there exists Z ∈ SB and a Borel function that assigns for each X ∈ B an isomorphic copy of X inside of Z (Theorem 5).

Uniformly factoring weakly compact operators and parametrised dualisation

“…The next lemma is [11,Lemma 3.7], and it will play an important role in Section 5. Recall that a critical ingredient towards showing that a separable Banach space embeds isometrically into (Δ) is the fact that * is separable and metrisable in the weak * topology and thus an image of the Cantor set under some continuous map.…”

“…The main result of this section is Theorem 3.1, which is a generalisation of [11,Theorem 1.1]. To prepare for the proof, we first introduce a coding for the unit ball of the duals of separable Banach spaces as compact subsets of the product space [−1, 1] N (we follow the approach of [16, Section 2.1.2]).…”

“…is Borel. Moreover, one can easily see that for each ∈ SB there exists a linear isometry [11,Section 3]).…”

“…As noticed by G. Godefroy in [20,Problem 5.2], the result [11,Theorem 1.1] implies that the equivalence class of a separable reflexive Banach space is Borel if and only if the equivalence class of its dual * is Borel. Another consequence of Theorem 3.1 is an analogous result for bounded operators, as we will see in Corollary 3.7.…”

“…Our main tool to prove Theorem B for A satisfying (2) or (3) therein is a new descriptive set-theoretic result (Theorem 3.1) that extends a theorem by the third named author (see [11,Theorem 1.1]). Precisely, let L SD denote the subset of L that codes all ofthe bounded operators :…”