11.3. Inner quasidiagonality 327Here is the main theorem of this section.
Theorem 11.3.9. The unital C*-algebra A is strong NF if and only if it's
separable unital nuclear and inner QD.Proof. If A is strong NF, then it is nuclear (Theorem 11.2.5) and an increas-
ing union of RFD subalgebras (Proposition 11.2.8). Corollary 11.3.7 then
implies inner QD because we can extend any homomorphism from an RFD
subalgebra to a c.c.p. map on all of A, by Arveson's Extension Theorem.
For the other direction we will appeal to the third condition in Theorem
11.2.7. So fix a finite set i C A, c > 0 and find some u.c.p. maps cp: A --+
Mn(C) and ~: Mn(C) --+ A such that Ila - ~(cp(a))ll < c for all a E i.
Applying Proposition 11.3.8 to cp, we can find a projection p in the socle
of A and a u.c.p. map 'lj.;: pAp --+ Mn(C) such that ll[a,pJll < c and
ll'lj;(pap) - cp(a)ll < c for all a E i.
Let Ap = An {p }' be the multiplicative domain of the compression map
A--+ pAp, let J <1 Ap be the kernel of this map (so pAp = Ap/ J = pAp)
and let {ei} CJ be a quasicentral approximate unit. Let B = pAp = pA**p
and I: B --+ Ap be a u.c.p. splitting for the quotient map Ap --+ B. It follows
that / is a complete order embedding. Thus the u.c.p. maps f3i: B --+ A
defined by
1 1 l - l
f3i(b) = (1-ei)21(b)(l - ei)2 + e[ f3('lj;(b))e[
are also complete order embeddings (since they're also splittings). We must
show that for large i, the images of these maps almost contain i.
If a E i, then we can find c E Ap such that Ila - ell < c. By quasicen-
trality we then have,
1 1 l l
a R:i c R:i (1-ei)2c(l - ei)2 + e[ce[.We also have lie - ~(cp(c))ll < 3c which implies that the distance between
1 1 l l
(1 - ei) 2 c(l - ei) 2 + ei^2 eel and
1 1 l - l
(1-ei)21(pc)(l - ei)2 +el f3('lj;(pc))e[
will be bounded above by 6c, for all large i, since 11 'lj.; (pc) - cp ( c) 11 < 3c. As
this latter element lives in the range of {3, the proof is complete. 0
We conclude with a striking decomposition theorem for simple nuclear
QD C* -algebras.
Corollary 11.3.10. If A is separable unital simple nuclear and QD, then
it contains an increasing sequence of nuclear RFD subalgebras whose union
is dense.