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Each concrete operator space is an operator space. The converse is given by
Ruan's theorem:
Each (abstract) operator space is completely isometrically isomorphic to a
concrete operator space [Rua88].
More concretely, for a matricially normed space
let
be the set of all complete
contractions from
to
. Then the mapping
is a completely isometric embedding of
into a
-algebra
[ER93].
A proof relies on the separation theorem for absolutely matrix convex sets.
This theorem can be used to show that many constructions with concrete operator spaces
yield again concrete operator spaces (up to complete isometry).
Prof. Gerd Wittstock
2001-01-07