Characterizations

For a matricially normed space $X$ the following conditions are equivalent:

a) $X$ is injective.

b) For each complete isometry $\iota:X\to Z$ there is a complete contraction $\pi:Z\to X$ such that $\pi\iota=\mathrm{id}_X$. I. e. $X$ is completely contractively projectable in each space containing it as a subspace.

c) For each complete isometry $\iota:X\to Z$ and each complete contraction $\varphi:X\to Y$ there is a complete contraction $\tilde{\varphi}:Z\to Y$ such that $\tilde{\varphi}\iota=\varphi$. I. e. Complete contractions from $X$ can be extended completely contractively to any space conaining $X$ as a subspace.¹⁹

d) $X$ is completely isometric to a completely contractively projectable subspace of $B(\H)$ for some Hilbert space $\H$.

e) $X$ is completely isometric to $pAq$, where $A$ is an injective $C^*$-algebra and $p$ and $q$ are projections in $A$.

Robertson characterized the infinite dimensional injective subspaces of $B(l_2)$ up to isometry (not complete isometry!). They are $B(l_2)$, $l_\infty$, $l_2$, $l_\infty\oplus l_2$ and $\bigoplus_{n\in{\mathbb{N}}}^{L^\infty}l_2$. (Countable $L^\infty$-direct sums of such are again comletely isometric to one of these.) If an injective subspace of $B(l_2)$ is isometric to $l_2$, it is completely isometric to ${\mathcal{R}}_{l_2}$ or ${\mathcal{C}}_{l_2}$.

Footnotes

... subspace.¹⁹

Equivalently: Completely bounded mappings from $X$ can be extended with the same norm.