On the complexity of some classes of Banach spaces and non-universality

Abstract

Abstract. These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded operators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks operators is complemented in its bounded operators, are all non Borel in SB. At last, we give several applications of those results to non-universality results.

Tsirelson-like spaces and complexity of classes of Banach spaces

“…In particular, these classes are not Σ 1 2 . This result answers two questions posed by B. M. Braga in [6]. We recall that a Banach space X is said to have the Schur property if every weakly convergent sequence in X is norm convergent.…”

“…Therefore, it follows from the Tsirelson space method that the class of all separable Banach spaces with the Schur property is not coanalytic. Using a tree space method developed in [3], it can be shown that this class is not analytic (see [6,Theorem 27]). Our proof of the proper complexity result (Theorem 1.3) can be considered as a combination of these two methods.…”

“…The corresponding set is analytic by [6,Theorem 20]. (e) It is easy to show that the corresponding set is Borel.…”

Amalgamations of classes of Banach spaces with a monotone basis

“…(V) If a separable Banach space X is isomorphically universal for separable Schur spaces, then it is actually isomorphically universal for all separable Banach spaces. This follows from methods in [3] (see [6,Corollary 51]). We are able to prove the isometric version of this statement (see Remark 3.7).…”

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…For each T ∈ Tr, let X 2,C[0,1],T be the metric space define above. An easy transfinite induction on the order of T gives us that, for all well-founded trees T ∈ Tr, the space X 2,C[0,1],T has the Banach-Saks property (see [Bra14], Theorem 14). Also, if T ∈ IF, it is clear that C[0, 1] linearly isometrically embeds into X 2,C[0,1],T .…”