1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

284 9. Local Reflexivity

the existence of a modular automorphism and semifiniteness of the corre-
sponding crossed product and then show how to derive what we need from
these important facts.

9.1. Local reflexivity

In Banach space theory, the principle of local reflexivity refers to the follow-
ing (cf. [120]):

Theorem 9.1.1. Let X be a Banach space and E c X** be a finite-

dimensional subspace. Then, there exists a net of contractions i.(Ji: E ---+ X

which converges to idE in the point-weak* topology. Moreover, we may as-

sume that the net { 1.fJi I EnX} converges in the point-norm topology to idEnx.

Being Banach spaces, C* -algebras thus enjoy the (Banach space) prin-

ciple of local reflexivity. But what if we require c.c.p. maps (instead of just

contractions)?

Definition 9.1.2. A C*-algebra A is locally reflexive if for every finite-

dimensional operator system E C A**, there exists a net of c.c.p. maps

i.(Ji: E---+ A which converges to idE in the point-ultraweak topology.

As usual, it is often convenient to reduce to the unital case.

Lemma 9.1.3. A nonunital C* -algebra A is locally reflexive if and only if

its unitization A is locally reflexive.

Proof. This is a special case of the next proposition. D

Proposition 9.1.4. Let 0---+ I---+ A---+ B ---+ 0 be short exact with A unital.

Then A is locally reflexive if and only if both I and B are locally reflexive

and the extension is locally split.

Proof. Let 7r: A ---+ B denote the quotient map. Suppose first that A is

locally reflexive and E c B** is a finite-dimensional operator system. By

the natural identification A** = I** E9 B**, we may regard E c A**. (If you

want an operator system, just consider E = E + ClA~*.) Since A is locally

reflexive, there exists a net of c.c.p. maps 'I/Ji: E ---+ A which converges to

idE in the point-O"(A**, A*) topology. Thus, the net 7r o 'I/Ji: E---+ B of c.c.p.

maps converges to idE in the point-O"(B**, B*) topology. This proves that B

is locally reflexive. We leave it to the reader to show the local reflexivity of I

(Exercise 9.1.1). To prove local liftability of 7r, we start with E c B (rather

than E C B**). Then the net 7r o 'I/Ji of liftable c.c.p. maps will converge to

idE in the point-weak topology; taking convex combinations, we can assume

norm convergence (see Lemma 2.3.4). Thus Arveson's Lemma 0.2 ensures

that idE is liftable too.