An operator space together with a completely isometric mapping is called an injective envelope of if is injective, and if is the unique extension of onto .
This is the case if and only if is the only injective subspace of which contains the image of .
Each matricially normed space has an injective envelope. It is unique up to a canonical isomorphism.
A matricially normed space together with a completely isometric mapping is called an essential extension of if a complete contraction is completely isometric if only is completely isometric.
is an injective envelope if and only if is injective and is an essential extension.
Every injective envelope is a maximal essential extension, i. e. for each essential extension , there is a completely isometric mapping such that .