[Paper Review] Remarks on Gurarii spaces

Joanna Garbulińska, Kubi\'s, Wies{\l}aw|arXiv (Cornell University)|Nov 24, 2011

Advanced Banach Space Theory4 citations

TL;DR

This paper investigates Gurariï spaces and their non-separable and strong variants, characterizing non-separable Gurariï spaces via skeletons of separable subspaces and constructing a non-separable Gurariï space with a projectional resolution of the identity. It proves that no strong Gurariï space can be weakly Lindelöf determined and shows that every copy of c₀ is complemented in Gurariï spaces built via pushout iterations.

ABSTRACT

We present selected known results and some of their improvements, involving Gurarii spaces. A Banach space is Gurarii if it has certain natural extension property for almost isometric embeddings of finite-dimensional spaces. Deleting the word "almost", we get the notion of a strong Gurarii space. There exists a unique (up to isometry) separable Gurarii space, however strong Gurarii spaces cannot be separable. The structure of the class of non-separable Gurarii spaces seems to be not very well understood. We discuss some of their properties and state some open questions. In particular, we characterize non-separable Gurarii spaces in terms of skeletons of separable subspaces, we construct a non-separable Gurarii space with a projectional resolution of the identity and we show that no strong Gurarii space can be weakly Lindel\"of determined.

Motivation & Objective

• . The paper aims to clarify the structure of non-separable Gurariï spaces and their relationship to separable subspaces.
• . It investigates the existence and properties of strong Gurariï spaces, particularly their incompatibility with weakly Lindelöf determined structures.
• . The authors seek to understand the role of c₀ in Gurariï spaces and its complementation properties under pushout constructions.
• . They explore the implications of rotund renorming and ultrahomogeneity for spaces of universal disposition.
• . The paper addresses open problems concerning universal disposition, homogeneity, and the existence of non-Euclidean classes of finite-dimensional spaces with amalgamation.

Proposed method

• . The authors use the pushout construction as a key technique to generate Gurariï spaces from finite-dimensional Banach spaces.
• . They characterize non-separable Gurariï spaces in terms of skeletons of separable subspaces, providing a structural criterion.
• . The proof of the projectional resolution of the identity (PRI) relies on constructing a transfinite sequence of separable subspaces with compatible projections.
• . The complementation of c₀ in pushout-constructed spaces is established using arguments from [1], showing that such spaces contain 1-complemented copies of c₀.
• . The paper applies forcing and absoluteness arguments to demonstrate that no space of universal disposition for separable spaces can admit a rotund renorming.
• . It employs Fraïssé-Jónsson limits and the theory of universal disposition to generalize the Gurariï property to larger classes of Banach spaces.

Experimental results

Research questions

• RQ1 . Can a non-separable Gurariï space with a projectional resolution of the identity be constructed?
• RQ2 . Is it possible for a strong Gurariï space to be weakly Lindelöf determined?
• RQ3 . Does every Gurariï space contain a complemented copy of c₀ when built via pushout iterations?
• RQ4 . Can a Banach space of universal disposition for separable spaces admit a rotund renorming?
• RQ5 . Is there a class K of finite-dimensional Banach spaces with the amalgamation property, not dense in the Banach-Mazur space, and not consisting of Euclidean spaces, that yields a non-Hilbertian, almost homogeneous space?

Key findings

• . A non-separable Gurariï space with a projectional resolution of the identity exists, demonstrating that such spaces can have rich geometric structure.
• . No strong Gurariï space can be weakly Lindelöf determined, indicating a fundamental structural limitation.
• . Every copy of c₀ is complemented in Gurariï spaces constructed via pushout iterations, a property that holds regardless of density.
• . The class of non-separable Gurariï spaces is characterized by the existence of a skeleton of separable subspaces satisfying specific extension properties.
• . There exist non-separable Gurariï spaces that admit a rotund renorming, showing that such geometric properties are not ruled out.
• . The paper shows that a space of universal disposition for separable spaces cannot admit a rotund renorming, using forcing and absoluteness arguments to establish this as a theorem independent of set-theoretic assumptions.