Amalgamations of classes of Banach spaces with a monotone basis

Abstract

ABSTRACT. It was proved by Argyros and Dodos that, for many classes C of separable Banach spaces which share some property P, there exists an isomorphically universal space that satisfies P as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that C consists of spaces with a monotone Schauder basis. For example, we prove that if C is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every X ∈ C is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for C.

Non-universal families of separable Banach spaces

“…The present paper, as well as two author's recent papers [18,19], establishes isometric counterparts of results concerning universality questions in separable Banach space theory and their natural connection with descriptive set theory (see [5,1,3,4,2,10,7,8,15,17]). These three papers together give a solution of a problem posed by G. Godefroy [14] if there exists any isometric version of the amalgamation theory of S. A. Argyros and P. Dodos [2] which would provide isometrically universal spaces for small, or regular, isometric classes of Banach spaces.…”

“…Nevertheless, there are still analogies between our methods and methods from [7]. The main analogy is that the result has been already proved for classes of spaces with a monotone basis (see [18,Theorem 1.2]) and so our task is to find an embedding of a general separable space into a space with a monotone basis. The embedding must preserve non-universality and must be simple from the descriptive set theoretic viewpoint.…”

“…We are going to apply the same technique as in [18,Section 8]. This will enable to obtain a new collection {|||·||| X : X ∈ SE (C([0, 1]))} of norms on Z with the same properties and with the additional property that all line segments contained in the unit sphere of (Z, |||·||| X ) are contained in JX.…”

Tsirelson-like spaces and complexity of classes of Banach spaces

“…Our second main result is based on a combination of methods used for proving Theorem 1.2 with a tree space method used in [20]. In particular, these classes are not Σ 1 2 .…”

“…In this section, we apply the construction of a tree space studied in [20] on Tsirelson type spaces presented above. This will enable us to show that some classes of Banach spaces have quite high complexity (Theorem 1.3).…”