Local reflexivity for C*-algebras - Imported Knowlpedia Content

Definition

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 Brown–Ozawa, §9.2.

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.