Ruan's theorem

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Ruan's theorem

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

$\displaystyle \Phi:X$ $\displaystyle \to$ $\displaystyle \bigoplus_{n\in{\mathbb{N}}}\bigoplus_{\Phi\in S_n}M_n$

$\displaystyle x$ $\displaystyle \mapsto$ $\displaystyle (\Phi(x))_{\Phi}$

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

$\displaystyle \Phi:X$	$\displaystyle \to$	$\displaystyle \bigoplus_{n\in{\mathbb{N}}}\bigoplus_{\Phi\in S_n}M_n$
$\displaystyle x$	$\displaystyle \mapsto$	$\displaystyle (\Phi(x))_{\Phi}$