Definition

A matricially normed space $X$ is called injective if completely bounded mappings into $X$ can be extended with the same norm. More exactly:

For all matricially normed spaces $Y_0$ and $Y$, each complete contraction $\varphi:Y_0\to X$ and each complete isometry $\iota:Y_0\to Y$ there is a complete contraction $\tilde{\varphi}:Y\to X$ such that $\tilde{\varphi}\iota=\varphi$.

It suffices to consider only operator spaces $Y_0$ and $Y$. Injective matricially normed spaces are automatically comlete, so they are also called injective operator spaces.