Banach spaces of universal disposition

TL;DR

This paper introduces a method to construct Banach spaces with universal disposition properties, including the Gurari space and Kubis space, and explores their structural and automorphic features.

Contribution

It provides a systematic method to obtain Banach spaces of universal and almost-universal disposition for various classes, and analyzes their properties and relationships.

Findings

Constructed the Gurari space and Kubis space using the method.

Showed Kubis space is not isomorphic to any subspace of a C(K)-space.

Under CH, proved Kubis space is isomorphic to an ultrapower of the Gurari space.

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.