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Exact operator spaces

We consider now the exact sequence

$\displaystyle 0\hookrightarrow
K(\ell_2) \hookrightarrow B(\ell_2) \to Q(\ell_2)\to 0.$

An operator space $ X$ is said to be exact [Pis95, §1], if the short sequence of injective tensor products

$\displaystyle 0\hookrightarrow
X\stackrel{\scriptscriptstyle \vee}{\otimes}K(\e...
...times}B(\ell_2) \to X\stackrel{\scriptscriptstyle \vee}{\otimes}Q(\ell_2)\to 0
$

is again exact. Then tensorizing with such an operator space preserves the exactness of arbitrary exact sequences of $ C^*$-algebras (for $ C^*$-algebras cf [Kir83]). Obviously, all finite dimensional operator spaces are exact.

Exactness is inherited by arbitrary subspaces. The injective tensor product of two exact operator spaces is again exact. For an exact space $ X$ we are given a degree of exactness by the quantity

$\displaystyle {\rm ex}(X)=\Vert X\stackrel{\scriptscriptstyle \vee}{\otimes}Q({...
...s}B({\cal H}))/(X\stackrel{\scriptscriptstyle \vee}{\otimes}K({\cal H}))\Vert. $

We have $ 1\leq{\rm ex}(X)<\infty$ [Pis95, §1], because the mapping

$\displaystyle (X\stackrel{\scriptscriptstyle \vee}{\otimes}B({\cal H}))/(X\stac...
...times}K({\cal H})
) \to X\stackrel{\scriptscriptstyle \vee}{\otimes}Q({\cal H})$

is a complete contraction. For non-exact operator spaces $ X$ we put ex $ (X)=\infty$.

For an exact $ C^*$-algebra 34 $ A$ we have ex$ (A)=1$.

For an operator space $ X$ we have:

$\displaystyle {\rm ex}(X)=\sup\{{\rm ex}(L):L\subset X , \dim L<\infty \}
.$

so we can confine our examinations to finite dimensional spaces. From this it is also immediate that: ex $ (X_0)\leq{\rm ex}(X)$ if $ X_0\subset X$. One has for finite dimensional operator spaces $ X_1$ and $ X_2$ the complete variant of the Banach-Mazur distance

$\displaystyle d_{CB}(X_1,X_2)=\inf\{\Vert\varphi\Vert _{{\rm cb}}
\Vert\varphi^{-1}
\Vert _{{\rm cb}}\}$

( the infimum is taken over all isomorphisms $ \varphi$ from $ X_1$ to $ X_2$ ). Via this Banach-Mazur distance we can define the quantity

$\displaystyle d_{SK}(X):=\inf\{d_{CB}(X,L), \dim (L)=\dim (X),
L\subset M_n , n\in {\mathbb{N}}\}
$

According to [Pis95, Thm. 1] ex $ (X)=d_{SK}(X)$ holds, and ex $ (X)\leq\sqrt{\dim(X)}$.



Footnotes

...-algebra 34
A characterization of exact $ C^*$-algebras is given in [Kir94] and [Kir95].

next up previous contents index
Next: Projective operator space tensor Up: Injective operator space tensor Previous: Some formulae for the   Contents   Index
Prof. Gerd Wittstock 2001-01-07