1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.4. Homotopy invariance 277

for all b EB, and

OC(JC) 0 I= UM2(0C(JC) 0 I)U*

where we identify Mz(OC(JC) 0 I) C IIB((JC 0 H) EB (JC 0 H)) canonically.

Corollary 8.4. 7. If 0 -* OC ® I -t E -t B -* 0 is exact, where OC ® I

is nuclear and essential in E, and there exists a *-homomorphic splitting

a-: B -* E, then for any faithful essential representation 1T: B -* IIB(JC)

there is an embedding of E into the algebra

1r(B) ®Cl + OC(JC) ®I.

Proof. Since OC ®I is essential in E, we may identify OC ®I~ OC(JC) Q9 I c

IIB(JC ® H), for some nondegenerate inclusion I c IIB(H), and hence regard

E c M(OC(JC) ® J) c IIB(JC ® H) and the splitting a- as taking values in

M(OC(JC) ® J).

Let lE: E <-t IIB(JC ® H) be the ·natural inclusion and extend the given

map 1T: B-* IIB(JC ® H) to Eby composing with the quotient map E-* B.

Evidently

lE EB 1r: E-* IIB((JC ® H) EB (JC® H))

is faithful and takes values in the C* -algebra

(o-EB 1r)(B) + Mz(OC(JC) ® J).

But the unitary from the previous theorem twists this algebra (isomorphi-

cally) over to the algebra 1r(B) ®Cl+ OC(JC) ®I, so we're done. D

Here is a trivial case of a general result, due to Larry Brown, which

asserts that extensions of AF algebras are always AF ([53, Theorem III.6.3]).

Lemma 8.4.8. Assume C is AF and 1T: C -* IIB(JC) is any representation.

Then 1T( C) + OC(JC) is also AF.

Proof. Let J c 1T( C) + OC(JC) be an arbitrary finite set and let c: > 0. By
density we may assume that each x E J can be written as x = 1T(cx) + Tx
where Tx E OC(JC) is a finite-rank operator. In this situation, we can find a
finite-rank projection P E OC(JC) such that PTx = TxP = Tx for all x E J.
Since C is AF, we can find a finite-dimensional subalgebra D C 1r(C)
which contains, within c:, all of the elements 1r(cx), x E J. Now let Q E OC(JC)
be the projection onto the subspace DPJC. Since this space is invariant for
D, Q commutes with D. Hence C*(D, QOC(JC)Q) = D + QOC(JC)Q is a finite-
dimensional subalgebra of 1r( C) + OC(JC) which nearly contains J. D

Putting things together, we arrive at (a special case of) a result of Spiel-
berg (cf. [179]).