Tsirelson-like spaces and complexity of classes of Banach spaces

Abstract

ABSTRACT. Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of c 0 is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the DunfordPettis property are Π 1 2 -complete.

The isomorphism class of C0 is not Borel

“…As noticed in [13], we obtain immediately that the class of isomorphic subspaces of c 0 is not Borel (that is the mentioned result from [20]), as the intersection of this class with the Borel class of all infinite-dimensional separable L ∞ -spaces is exactly the isomorphism class of c 0 (see [18,Corollary 1]). This works in one direction only.…”

“…Godefroy asked in [13] if the isomorphism class of the space c 0 is Borel. In the preceding paper [20], we have found a partial solution, proving that the class of all spaces isomorphic to a subspace of c 0 is not Borel. The main result of the present work is a full solution of Godefroy's problem.…”

A New Coarsely Rigid Class of Banach Spaces

“…is also analytic. This is because a Suslin operation of analytic sets is an analytic set (see, e.g., [ We now show that the class is R ∩ As c 0 is not analytic and hence by Suslin's theorem (see, e.g., [ [25] also provides a proof of the fact that R ∩ As c 0 is not Borel. b) It also follows easily from [32, Theorem 1.2] that given 1 < p < ∞ the class of all separable reflexive spaces that are asymptotic ℓ p is analytic.…”

The complexity of some ordinal determined classes of operators

“…. Now for T ∈ Tr(2 × N), we define the space E T to be the completion of c 00 (2 <N \ {∅}) with respect to the norm Here, for a compact set M ⊂ 2 N , · M denotes the Tsirelson space T * [M, 1/2] as defined in [19]. Kurka showed that there exist Borel maps S, S * : Tr(2 × N) → SB such that for each T ∈ Tr(2 × N), S(T ) is isometric to E T and S * (T ) is isometric to E * T .…”