Characteristic properties of the Gurariy space

Abstract

The Gurariy space G is defined by the property that for every pair of finite dimensional Banach spaces L ⊂ M, every isometry T: L → G admits an extension to an isomorphism \(\mathop T\limits^ \sim :M \to G\) with ‖T‖‖T−1‖ ≤ 1 + ∈. We investigate the question when we can take \(\mathop T\limits^ \sim \) to be also an isometry (i.e., ∈ = 0). We identify a natural class of pairs L ⊂ M such that the above property for this class with ∈ = 0 characterises the Gurariy space among all separable Banach spaces. We also show that the Gurariy space G is the only Lindenstrauss space such that its finite-dimensional smooth subspaces are dense in all subspaces.