1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3. 
   
   
   
   7. Exact sequences 95
   
   
   
   

Lemma 3. 7. 7. If J <l A is an ideal and B C C is a C*-subalgebra, then

( J 0 C) n (A 0 B) = J @ B.

Proof. The inclusion :J is immediate, so assume that x E (J@C)n(A@B).

Let {en} C J be an approximate unit and let us assume for the moment that

C is unital. Then {en @ lo} C J @ C is an approximate unit (use Lemma

3.4.10) and hence the elements (en@lo)x belong to J@B (since x E A@B)

and converge in norm to x (since x E J@C). But .J@B is norm closed and

so we are finished in this case.

If C is not unital, then we can unitize and apply the argument above. D

Proposition 3.7.8. If B is exact, then

0--+ J 0 B--+ A 0 B--+ (A/ J) 0 B--+ 0

is exact for all A and J <l A.^21

Proof. If J <l A is arbitrary and an element x E A@ B is in the kernel of
A @ B --+ (A/ J) @ B, then we must convince ourselves that x E J @ B.

Let 7r: B '--+ lffi(H) be a faithful representation and find some c.c.p. maps
<pn: B--+ Mk(n)(C), 1/Jn: Mk(n)(C)--+ lffi(H) such that 117r(b)-1/Jno'Pn(b)ll--+ 0
for all b E B. Consider the following commutative diagram:

0 ----+ J@B ----+ A@B ----+ (A/J)@B ----+^0

id0<pn 1 id0<pn 1 id@<pn^1

0 ----+ J@Mk(n)(C) ----+ A@Mk(n)(C) ----+ (A/ J) 0 Mk(n) (C) ----+ 0

id0'¢n 1 id0'¢n 1 id0'¢n 1

0 ----+ J @lffi(1i) ----+ A@lffi(1i) ----+ (A/ J)^0 lffi(H) ----+

Since the middle row is exact, a bit of diagram chasing shows that

id@ (1/Jn o 'Pn)(x) E J@ B(H)

for all n. Since llid@7r(x) - id@ (1/Jn o 'Pn)(x)ll--+ 0, this shows that

id@ 7r(x) E (J @lffi(1i)) n (A@ 7r(B)).

By the previous lemma, this completes the proof since

id@7r: J@B--+ J 0 7r(B) = (J@lffi(1i)) n (A@7r(B))

is an isomorphism.

0.

D

21This actually characterizes exact C*-algebras and explains the terminology (see Theorem
3.9.1).