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
.