1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

324 11. Simple C* -Algebras

At the other end of the spectrum, if A is simple and QD, then it is

necessarily inner QD as well. Indeed, we can fix an irreducible representation

A c IIB(H) - which is necessarily faithful - and then the representation

theorem (Theorem 7.2.5) provides increasing, asymptotically commuting,

finite-rank projections on 1{.^7 Irreducibility ensures that these projections

live in the weak closure of Ac IIB(H) and thus give rise to projections in the

socle of A**.

Remark 11.3.4. Assume p EA** is a projection in the socle and let c(p)

be the central cover. Then

c(p )A** Co! IIB(H1) EB · · · EB IIB(Hn)

for some Hilbert spaces H1, ... , Hn· That c(p)A** contains no summand

of type II or III follows from the existence of minimal projections (pA**p

contains minimal projections) and then the result follows from the struc-

ture theory of type I von Neumann algebras. (If p E L^00 (X, μ) ® IIB(H) ~

L^00 ((X,μ),IIB(H)) has cental cover 1 and lives in the socle, thenμ must be

a finite sum of point masses.)

Though it isn't obvious, the benefit of inner quasidiagonality is that the

c.p. maps in the definition of QD can be taken to have large multiplicative

domains. This requires a lemma.

Lemma 11.3.5. Let p EA** be a· projection in the socle and define

Ap ={a EA: [a,p] = O}.

The;i p belongs to the weak closure of Ap in A** (i.e., A;*) and, hence, is

the central cover of the *-representation Ap -+ pAp.

Proof. Fix an identification

c(p )A** ~ IIB(H1) EB · · · EB IIB(Hn),

where c(p) is the central cover (in A**) of p. Applying (a tiny modification

of the proof of) Corollary 1.4.8, we can find a net of self-adjoints ai E A such

that ai -+ p ultraweakly and aip = p for all i. Taking adjoints, it follows

that aip = pai and hence p E A;* as claimed.

It is immediate from the definition that p must be the central cover of

the homomorphism Ap-+ pAp, so the proof is complete. D

Proposition 11.3.6. Let p E A** be in the socle and let Ap = {a E A :

[a,p] = O}, as above. Then d(a,Ap) = li[a,p]ll for all a E A.^8

(^7) Note that if A is not isomorphic to K, then the representation A c lffi(7i) is necessarily
essential -by simplicity -whereas the case A = K is trivial.
(^8) Note that Ap is the multiplicative domain of the compression map a f--+ pap.