Banach spaces of universal disposition

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurari˘ı space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurari˘ı space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness. © 2011 Elsevier Inc. All rights reserved.