Exact operator spaces

We consider now the exact sequence

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

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

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

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

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

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

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

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:

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

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

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

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

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

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].