Banach spaces of universal disposition

Abstract

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class $\mathfrak M$ of normed spaces. The method produces, among other, the Gurari\u{\i} space $\mathcal G$ (the only separable Banach space of almost-universal disposition with respect to the class $\mathfrak F$ of finite dimensional spaces), or the Kubis space $\mathcal K$ (under {\sf CH}, the only Banach space with the density character the continuum which is of universal disposition with respect to the class $\mathfrak S$ of separable spaces). We moreover show that $\mathcal K$ is not isomorphic to a subspace of any $C(K)$-space -- which provides a partial answer to the injective space problem-- and that --under {\sf CH}-- it is isomorphic to an ultrapower of the Gurari\u{\i} space. We study further properties of spaces of universal disposition: separable injectivity, partially automorphic character and uniqueness properties

Universal disposition is not a 3-space property

“…Gurariy conjectured the existence of Banach spaces of universal disposition, as well as of spaces of universal disposition with respect to the class S of separable spaces. This conjecture was proved to be true in [1], where a general method to construct spaces of universal disposition with respect to different classes was presented. In particular, it was shown that the space that Gurariy conjectured is isometric to the Fraïssé limit, in the category of separable Banach spaces and into isometries, constructed by Kubiś [22].…”

$\aleph$-injective Banach spaces and $\aleph$-projective compacta

“…The push-out and pull-back constructions. A thorough description of the pull-back and push-out constructions in Banach spaces can be seen in [4,3,9]. Everything we need to know for this paper is that given an exact sequence (1) and an operator t : Y → B there is a commutative diagram…”

“…It can be proved (see [11,Theorem 6.1.8] or [12,Theorem 13.9]) that if U is countably incomplete and ℵ-good, then every family of less than ℵ internal subsets of S i U having the finite intersection property has nonempty intersection. Since |Γ × N| < ℵ and U is ℵ-good, there is x ∈ X i U in the nonempty intersection A combination of [4] and [3], see also [5], shows that there exist universally 1-separably injective spaces not isomorphic to any C(K) space. A higher cardinal generalization is as follows.…”

Banach Spaces of Almost Universal Complemented Disposition

“…The paper [2] started the study of spaces of universal disposition after Gurariy [10]. All available results and several new were then presented in Chapter 4 of the monograph [4].…”

“…• Spaces of S-universal disposition are 1-separably injective [2] and [4,Thm. 3.5]. • There are spaces of F-universal disposition that are not separably injective.…”