Remarks on Gurariĭ spaces (Q2906824)

For the reader who wants to enter the theory of Gurarii spaces, this article is definitely the place to start as it is written in an informative style, contains important historical remarks on the development of this theory as well as presenting the updated theory of Gurarii spaces. Let us give a hint of what the article contains, but first let us recall some facts:NEWLINENEWLINE\noindent 1. A Banach space \(X\) is of (almost) universal disposition for a class \(\mathcal{K}\) of Banach spaces if, for every pair \(E\subset F\) from \(\mathcal{K}\), every isometric embedding \(i:E\to X\) can be extended to an (almost) isometric embedding \(j:F\to X\).NEWLINENEWLINE\noindent 2. Gurarii himself constructed a separable Banach space \(G\) of almost universal disposition for the class of finite-dimensional Banach spaces [\textit{V. I. Gurarij}, Sib. Mat. Zh. 7, 1002--1013 (1966; Zbl 0166.39303)]. Any Banach space \(X\) of almost universal disposition for the class of finite-dimensional Banach spaces is called a Gurarii space. If \(X\) should happen to be of universal disposition for the class of finite-dimensional Banach spaces, then \(X\) is called strong Gurarii.NEWLINENEWLINE\noindent 3. \textit{W. Lusky} proved in [Arch. Math. 27, No.~6, 627--635 (1976; Zbl 0338.46023)] that the separable \(G\) constructed by Gurarii is unique up to isometrical isomorphism. Moreover, it is known that \(G\) is isometrically universal for all separable Banach spaces (contains an isometric copy of any separable Banach space). \(G\) is a Lindenstrauss space (the predual of an \(L_1(\mu)\)-space) and \textit{P. Wojtaszczyk} proved in [Stud. Math. 41, 207--210 (1972; Zbl 0233.46024)] that any separable Lindenstrauss space is isometric to a 1-complemented subspace of \(G\).NEWLINENEWLINE\noindent 4. A skeleton in a Banach space \(X\) is a directed set (w.r.t.\ set inclusion) of separable subspaces of \(X\) for which the union is all of \(X\). The intersection of two skeletons is again a skeleton (Proposition 3.1). A projectional resolution of the identity (PRI) in \(X\) is a transfinite sequence of norm 1 projections \((P_\alpha)_{\alpha<\omega_1}\) whose images are separable, form a continuous chain covering \(X\) and \(P_\alpha P_\beta =P_{\min(\alpha,\beta)}\) for all \(\alpha,\beta<\omega_1\).NEWLINENEWLINESection 1 of the paper contains definitions and explains the idea of the pushout technique. In Section 2, a short proof of Lusky's result is given, and then two ways of constructing \(G\) are presented, the simplest using the pushout technique, before the section ends with a proof of Wojtaszczyk's result.NEWLINENEWLINEIn Section 3, skeletons are introduced and it is proved that a Banach space is a Gurarii space if and only if it has a skeleton of spaces isometrically isomorphic to \(G\) (*). It follows that no complemented subspace of a \(C(K)\)-space is Gurarii. Using the pushout technique in tandem with (*), the authors now prove that every Banach space \(X\) embeds isometrically into a Gurarii space of the same density. Section 3 ends with a proof that there exists a Gurarii space of density \(\aleph_1\) and an argument that this indeed implies the existence of at least two non-isomorphic Gurarii spaces of density \(\aleph_1\).NEWLINENEWLINEThe theme of Section 4 are spaces of universal disposition for larger classes than the finite-dimensional spaces. First, the connection between the terms universal disposition and isometrical universality is presented and it is proved that there exists a space of density \(\mathfrak{c}\) of universal disposition for separable spaces, implying the existence of strong Gurarii spaces. Knowing that strong Gurarii spaces exist, the issue of Section 5 is to describe some of their structure. Gurarii [loc.\,cit.]\ already observed that such spaces can never be separable. The authors here prove that a strong Gurarii space is never 1-injective for finite-dimensional spaces and that in every strong Gurarii space one may pick a skeleton containing an element which is not 1-complemented. The last result implies that a strong Gurarii space can never be weakly Lindelöf determined, in particular, never WCG.NEWLINENEWLINEIn Section 6, Sobczyk's lemma is extended to a class of spaces which include some strong Gurarii spaces. Again, the pushout technique is fundamental. Section 7 is a list of open questions, many of them naturally arising from the work of the previous sections.