Characteristic properties of the Gurariy space
A Gurariy space is a Banach space \(Z\) with the following property: Given finite-dimensional Banach spaces \(X\subset Y\), \(\varepsilon>0\) and a linear isometric embedding \(f:X\to Z\), then \(f\) extends to a linear mapping \(g:Y\to Z\) with \(\|g\|\cdot\|g^{-1}\|<1+\varepsilon\) (an \(\varepsilon\)-isometry into). These spaces are named after \textit{V. I. Gurarij} who constructed in [Sib. Mat. Zh. 7, 1002--1013 (1966; Zbl 0166.39303)] a separable such Banach space, \(G\). \textit{W. Lusky} proved in [Arch. Math. 27, 627--635 (1976; Zbl 0338.46023)], using deep techniques of \(L_1\)-preduals due to Lazar and Lindenstrauss, that \(G\) is unique among separable spaces. The authors prove two ``\(\varepsilon\)-free'' characterizations of \(G\). We present here the one of them that we find particularly nice and also easy to remember: A separable Banach space \(X\) is \(G\) if and only if: \noindent (a) Whenever \((L,M)\) is a pair of finite-dimensional spaces with \(\dim(M/L)=1\) and such that Hahn-Banach extensions from \(L\) to \(M\) are unique, we may extend any linear isometry from \(L\) into \(X\) to a linear isometry from \(M\) into \(X\). Both directions of the above characterization are very involved. In the last part of the paper, \(G\) is characterized among Lindenstrauss spaces (= \(L_1\)-preduals) in a completely different way: \noindent (b) \(G\) is the only Lindenstrauss space for which the family of smooth finite-dimensional subspaces is dense in the family of all finite-dimensional subspaces. Here, denseness is in terms of the complete metric \(\theta\) on the set of closed subspaces of a Banach space \(X\) given by \[ \theta(L,M)=\max\{\sup_{x\in B_L}d(x,B_M),\sup_{x\in B_M}d(x,B_L)\}. \]
scientific article; zbMATH DE number 6375678