318 11. Simple C*-AlgebrasRemark 11.2.2. If a u.c.p. map r.p: A -+ E is a complete order embedding,
then r.p-^1 : r.p(A)-+ A is also completely positive -i.e., r.p is an isomorphism,
in the category of operator systems, onto its range (hence the terminology).
Indeed, any unital completely contractive map is completely positive ([63,
Corollary 5.1.2]), so in particular this is true for r.p-^1.
However, more is true: r.p-^1 extends to a (surjective) *-homomorphism
C*(r.p(A))-+ A. To see this, we assume Ac JE(H) and extend r.p-^1 to a u.c.p.
map E -+JECH). The point is that r.p(A) is contained in the multiplicative
domain of this map, since 1 ~ r.p-^1 ( r.p( u )*r.p( u)) ~ r.p-^1 ( r.p( u )*)r.p-^1 ( r.p( u)) = 1,
for all unitaries u E A (and similarly with adjoints on the right).Definition 11.2.3. Let A = g lim(Am, 1Pn m) be a generalized inductive
----> '
limit where each Am is finite-dimensional. If the connecting maps 1Pn,m are
c.c.p. then we say A is an NF algebra.^4 If the 1Pn,m's are all complete order
embeddings, then A is strong NF.Though it isn't obvious, NF algebras are always nuclear.
Proposition 11.2.4. NF algebras are nuclear and QD.Proof. First let A = g lim(Am, ____, 1Pn ' m) be a strong NF algebra with con-
necting maps 1Pn,m· If <I>m: Am -+ A are the canonical maps, then ev-
idently each <I>m is a complete order embedding of Am into A. Hence
<I>;:;/: <I>m(Am) -+ Am can be extended to a u.c.p. map A -+ Am· Evidently
this implies A is nuclear and QD.
For the general case, we may assume everything is unital. Indeed, if
A = g lim(Am, 1Pn m) and the 1Pn m's are not unital, then we adjoin new
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units everywhere and one checks that A = glim(Am,<Pnm)· ----> , Since A is
nuclear and QD whenever A is, our reduction is complete.
So, assume A= glim( Am, 1Pn m) with u.c.p. connecting maps. We have
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to use the strong NF case handled above. Indeed, we consider the generalized
inductive system (Em, 'l/Jn,m) where
Em = Ai EB · · · EB Am
and
'l/Jn,m(a1 EB··· EB am)= al EB··· EB am EB 1Pm+1,m(am) EB··· EB i.pn,m(am)·
Since the 'l/Jn,m's are complete order embeddings, E = glim(Em,'l/Jn,m) is
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nuclear. It is not hard to see that A is a quotient of E since we have a(^4) This abbreviates "nuclear and (stably) finite,'' even though it isn't known whether or not
every nuclear stably finite C* -algebra is NF.