Local reflexivity for C*-algebras

A $C^*$ -algebra $A$ is locally reflexive if, whenever $E\subseteq A^{**}$ and $F\subseteq A^*$ are finite-dimensional and $\varepsilon>0$ , there is a linear map $\phi:E\to A$ such that

\|\phi\|_{\mathrm{cb}}\leq 1+\varepsilon,\qquad f(\phi(x))=x(f)\quad(x\in E,\ f\in F),

and $\phi(x)=x$ for $x\in E\cap A$ . Here $\|\cdot\|_{\mathrm{cb}}$ is the completely bounded norm. The property says that finite-dimensional operator-space data in the bidual can be realized almost isometrically in $A$ , while matching any prescribed finite collection of dual pairings. No global bounded projection from $A^{**}$ onto $A$ is asserted by this definition.

Approximation formulation

Equivalently, for each finite-dimensional $E\subseteq A^{**}$ , the inclusion $E\hookrightarrow A^{**}$ is a point weak-star limit of maps $E\to A$ whose completely bounded norms tend to $1$ , with the maps fixing $E\cap A$ . Passing between this net formulation and the finite set $F\subseteq A^*$ formulation is a finite-dimensional separation argument. The matrix norm is essential: ordinary Banach-space local reflexivity holds for every Banach space, while operator-space local reflexivity is a genuine restriction.

Relation to exactness

Every exact $C^*$ -algebra is locally reflexive. In particular, nuclear $C^*$ -algebras are locally reflexive. Local reflexivity is nevertheless weaker than exactness; it controls finite-dimensional approximation from the bidual rather than the behavior of all short exact sequences under minimal tensor product.

One tensorial formulation says that the natural comparison maps involving finite-dimensional operator spaces and $A^{**}$ preserve the minimal operator-space norm. Such formulations make local reflexivity useful in passing approximation and lifting arguments between $A$ , its representations, and its weak closures.

Conventions

Some sources define local reflexivity for an arbitrary operator space $X$ and then apply that definition to the canonical operator-space structure of a $C^*$ -algebra. The interpolation requirement $\phi|_{E\cap A}=\operatorname{id}$ may be omitted from an equivalent version; it can be restored without changing the property.

References

Edward G. Effros and Uffe Haagerup, “Lifting problems and local reflexivity for $C^*$ -algebras,” Duke Mathematical Journal 52 (1985), 103–128. Project Euclid DOI record. Relevant: local reflexivity and its relation to lifting and tensor products.
Nathanial P. Brown and Narutaka Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. AMS DOI record. Relevant: §9.2 on local reflexivity and exact $C^*$ -algebras.