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\uı 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\uı space. We study further properties of spaces of universal disposition: separable injectivity, partially automorphic character and uniqueness properties.