Bounded injectivity and Haagerup tensor product (Q2843833)

An operator space \(V\) is said to be \textit{\(b\)-injective} if there is \(\lambda\geq 1\) such that, for given operator spaces \(W_{1}\subseteq W_{2}\), any completely bounded map \(\varphi_{1}: W_{1}\rightarrow V\) can be extended to a completely bounded map \(\varphi_{2}: W_{2}\rightarrow V\) with \(\|\varphi_{2}\|_{cb}\leq\lambda\|\varphi_{2}\|_{cb}\).NEWLINENEWLINEIn this paper, the authors prove a necessary and sufficient condition for an operator space \(V\) to be \(b\)-injective, \(V\) being completely isomorphic to a second operator space \(W\), and use the notions of injective and Haagerup tensor products as well as infinite matrices of operator spaces, related to notations and theorems which can be found in [\textit{D. P. Blecher} and \textit{C. Le Merdy}, Operator algebras and their modules -- an operator space approach. Oxford: Oxford University Press (2004; Zbl 1061.47002)] and [\textit{E. G. Effros} and \textit{Z.-J. Ruan}, Operator spaces. Oxford: Clarendon Press (2000; Zbl 0969.46002)]. The main result is to prove, for \(V\subseteq B(H)\) being an injective operator system on a separable Hilbert space \(H\), that \(V\otimes_{h}W\) is \(b\)-injective for all injective operator space \(W\) if and only if \(V\) is finite dimensional.