1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

322 11. Simple C*-Algebras

that 114? -1/illcb =max lllfz -1/izllcb < c:. Moreover, 1,0 is completely isometric
since the map A ----+ A defined by

A :3 x 1---t EB EB QkUzVz1/i(x)Vz*Uz*Qk E EB EB lBl(C~(k) l8l C'fJk) ~A

l kEKi l kEKz

can be shown to be a *-isomorphism.^0

Theorem 11.2. 7. For a unital separable C -algebra A, the following are
equivalent:
(1) A is strong NF;
(2) for every finite set i c A and c: > 0 there exists a finite-dimensional
C-algebra B and u.c.p. maps a: A ----+ B, {3: B ----+ A such that
llf3(a(a))-all < c: for all a E i and {3 is a complete order embedding;
(3) for every finite set i c A and c: > 0 there exist a finite-dimensional
C* -algebra B and complete order embedding {3: B ----+ A such that
i C^6 {3(B) (i.e., for each a E i there exists b E B such that
Ila - f3(b)ll < c:J.

Proof. Evidently we only have to prove (2) =?-(1). But, with the previous
perturbation result in hand, the proof is very similar to (4) :::::}- (1) from
Theorem 11.2.5, so we leave the details to the reader. 0

While we can certainly give lots of examples of NF algebras (e.g., the
cone over any nuclear algebra), we will have to wait until the next section
to see examples of strong NF algebras; quite remarkably they are abundant.
In the meantime, let's observe that they always admit a nice inductive limit
structure.

Proposition 11.2.8. If A is strong NF, then there exist subalgebras Ar C
A2 C · · · CA such that each An is nuclear, RFD and the union of the An 's
is dense in A.

Proof. If A = glim( ---+ Bm, <pn , m) where each Bm is finite-dimensional and the

connecting maps are all complete order embeddings, then it is not hard to see
that the subalgebra C(m(Bm)) c A is residually finite-dimensional. (Any
completely positive extensions of the maps ~^1 : n(Bn) ----+ Bn, n > m, will
contain C(m(Bm)) in their multiplicative domains.) Unfortunately there
is no reason to believe that C*(m(Bm)) is nuclear.