Tsirelson like operator spaces (Q2862561)

Let \(H\) be a separable, infinite dimensional and perfectly Hilbertian operator space. Weak-\(H\) operator spaces were recently introduced as an analogue of weak Hilbert Banach spaces in [\textit{H. H. Lee}, Houston J. Math. 35, No. 4, 1171--1201 (2009; Zbl 1187.47058)]. It was also shown there that an operator space \(E\) is a weak-\(H\) space if and only if, for any \(0 < \delta < 1\), there is a constant \(C > 0\) such that, for any finite dimensional \(F \subseteq E\), we can find \(F_1 \subseteq F\) and an onto projection \(P: E \to F_1\) satisfying NEWLINE\[NEWLINE d_{cb}(F_1,H_{\dim F_1}) \leq C, \quad \dim F_1 \geq \delta \dim F \quad \text{and} \quad \|P\|_{cb} \leq C, NEWLINE\]NEWLINE where \(d_{cb}\) is the completely bounded distance.NEWLINENEWLINELet \(C_p = [C,R]_{\frac{1}{p}}\) be the interpolation spaces of the column and the row Hilbert spaces via the complex method (\(1 \leq p \leq \infty\)). Non-trivial examples \(X_{C_p}\) of weak-\(C_p\) are constructed. These operator spaces have a local operator space structure very close to \(C_p\) and are non-homogeneous Hilbertian operator spaces that are not completely isomorphic to \(C_p\). The construction is inspired by W.~B.~Johnson's construction of \(2\)-convexified Tsirelson space.