1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

238 7. Quasidiagonal C* -Algebras

Remark 7.1.2. Completely positive maps respect linear, involutive and

order structures; the definition above requires that all other C* -structures

(i.e., norm and multiplication) be asymptotically preserved. Hence, one can

think of QD C* -algebras as those which admit matrix models approximately

recapturing all C* -structures.^1

Just as for exactness and nuclearity, this is really a local property (hence,

there is no substantial difference between the separable and nonseparable

settings). Obviously we can localize as follows:

Lemma 7.1.3. A is QD if and only if for each finite set i c A and c > 0

there exists a c.c.p. map cp: A-> Mn(<C) such that

llcp(ab) - cp(a)cp(b)ll < c

and

llcp(a)ll > llall - c

for all a, b E ~.

As always, there are some nonunital questions that need to be resolved.

Lemma 7.1.4. If A is unital and QD, then there exist u.c.p. maps cpn: A---->

Mk(n) (<C) which are both asymptotically multiplicative and asymptotically

isometric.

Proof. Let cp~: A----> Mz(n) (<C) be asymptotically multiplicative and isomet-

ric c.c.p. maps. Functional calculus shows that the spectra of the matrices

cp~(lA) are contained in sets of the form [O, en) U (en, 1], where En----> 0, and

hence

llcp~(lA) - Pnll ----> 0,

where Pn E Mz(n)(<C) are the spectral projections of cp~(lA) corresponding

to [1/2, l].

Thus cp~ (lA)Pn is an invertible element in PnMz(n) (<C)Pn and more func-

tional calculus shows

1

ll(cp~(lA)Pn)-2 - Pnll----> 0

as well. If k(n) is the rank of Pn, then we get the desired u.c.p. maps

!.pn: A-> Mk(n)(<C) by defining

1 1

cpn(a) = (cp~(lA)Pn)-2cp~(a)(cp~(lA)Pn)-2.

D

(^1) This sentence better describes MF algebras (see Definition 11.1.6). For those who care, the
thing which distinguishes QD algebras from MF algebras is the existence of c.c.p. maps connecting
the algebra with its matrix models.