The non-commutative Gurarii space (Q2492892)

A separable exact operator space \(X\) is called a noncommutative Gurariĭ space if it satisfies the following property: For any finite-dimensional exact operator space \(F\), for any \(E\subset F\), for any \(E'\subset X\), for any isomorphism \(u:E\to E'\) and for any \(\delta>0\), there exists a subspace \(F'\subset X\) containing \(E'\) and a linear isomorphism \(v\colon F\to F'\) extending \(u\), such that \[ \| v\|_{\text{cb}}\| v^{-1}\|_{\text{cb}}<(1+\delta)\| u\|_{\text{cb}}\| u^{-1}\|_{\text{cb}}. \] The main result asserts that if \(Y\) is a separable \(\mathcal{OL}_{\infty,1+}\) space in Effros-Ruan's sense, there exists a noncommutative Gurariĭ space \(X\) which contains \(Y\) as a completely contractively complemented subspace. On the other hand, any noncommutative Gurariĭ space is an \(\mathcal{OL}_{\infty,1+}\) space.