Complete Local Reflexivity

An operator space $X$ is called completely locally reflexive [EJR98, §1], if to each finite dimensional subspace there is $L \subset X ^{**}$ a net of completely contractive mappings $\varphi_\alpha : L\rightarrow X$ that converges to the embedding $L \rightarrow X^{* *}$ in the point weak$^*$ topology.⁵³

This property is inherited by arbitrary subspaces. In general it is not preserved by quotients. In the case that for example the kernel is an M-ideal (e.g. the twosided ideal of a $C^*$-algebra) and that the original space is completely reflexive we have that the quotient space is completely locally reflexive [ER94, Thm. 4.6].

Banach spaces are always locally reflexive (Principle of local reflexivity [Sch70]).

On the contrary, not all operator spaces are completely locally reflexive. For example the full $C^*$-algebra of the free group on two generators $C^*(F_2)$ and $B(\ell_2)$ are not completely locally reflexive [EH85, p. 124-125].

An operator space $X$ is completely locally reflexive, if and only if one (and then every) of the following conditions is satisfied for all finite dimensional operator spaces $L$ [EJR98, §1, 4.4, 5.8]:

1. $L\stackrel{\scriptscriptstyle \vee}{\otimes}X^{**}\stackrel{\mathrm{cb}}{=}(L\stackrel{\scriptscriptstyle \vee}{\otimes}X)^{**}$, where $\stackrel{\scriptscriptstyle \vee}{\otimes}$ denotes the injective operator space tensor product,
2. $\mathit{CB}(L^*,X^{**})\stackrel{\mathrm{cb}}{=}\mathit{CB}(L^*,X)^{**}$,
3. $L^*\stackrel{\scriptscriptstyle \wedge}{\otimes}X^*\stackrel{\mathrm{cb}}{=}(L\stackrel{\scriptscriptstyle \vee}{\otimes}X)^*$, where $\stackrel{\scriptscriptstyle \wedge}{\otimes}$ denotes the projective operator space tensor product and $\stackrel{\scriptscriptstyle \vee}{\otimes}$ denotes the injective operator space tensor product,
4. $\mathit{CN}(X,L^*)\stackrel{\mathrm{cb}}{=}\mathit{CI}(X,L^*)$, where $\mathit{CN}(\cdot,\cdot)$ denotes the completely nuclear and $\mathit{CI}(\cdot,\cdot)$ the completely integral mappings,
5. $\iota(\varphi)=\iota(\varphi^*)$ for all $\varphi\in CI(X,L^*)$.

In the conditions 1), 2) and 3) it suffices to check the usual isometry to prove the complete isometry.

Footnotes

... topology.⁵³

The difference to the definition of local reflexivity is the fact that the $\varphi_\alpha$ are not only supposed to be contractive, but even completely contractive.