On Banach spaces of universal disposition
Let \(\mathcal M\) denote a class of Banach spaces. A Banach space \(X\) is said to be of universal disposition for \(\mathcal M\) if, given \(A\) and \(B\) in \(\mathcal M\) and linear isometries \(u:A\to X\) and \(i:A\to B\), there exists a linear isometry \(v:B\to X\) satisfying \(u=v\circ i\). It has been proved in Theorem 3.1 of \textit{A. Avilés} et al. [Separably injective Banach spaces. Cham: Springer (2016; Zbl 1379.46002)] that each space of universal disposition for the class of separable spaces is separably injective. In the paper under review, it is proved that there is a space \(Y\) of universal disposition for the class of finite-dimensional spaces that contains complemented copies of all isometric preduals of \(\ell_1\), and a space \(Z\) of universal disposition for the class of finite-dimensional polyhedral spaces that contains complemented copies of \(C[0,1]\) and has the separable complementation property. Since every infinite-dimensional separable complemented subspace of a separably injective space is isomorphic to \(c_0\), the spaces \(Y\) and \(Z\) are not separably injective. In Section 6.6 of the previously mentioned reference, it has been conjectured that there is a continuum of mutually non-isomorphic spaces with density character \(\mathfrak c\) which are of universal disposition for the class of finite-dimensional spaces. Here, a new type of space of this kind is constructed.
scientific article; zbMATH DE number 6613972
scientific article