1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

7. 
   
   
   
   3. Homotopy invariance 251
   
   
   
   

down the commutator in matrix form, you will see that one must estimate

the norms of elements of the form

CYj_ ( ) b Gj 1/2 1/2 GJ+ 1 - Gj 1/2 GJ+1/2 1 CYj+i (b).

n n

However, this operator has small norm since CY j_ (b) almost commutes with

1/2 1/2 · n.

G j and G j+l and, furthermore, the norm of CY j_ (b) - CY J+1 (b) is small. D

n n

Theorem 7.3.6 (Voiculescu). If A homotopically dominates B and A is

QD, then B is also QD. In particular, quasidiagonality is a homotopy-

equivalence invariant.

Proof. Let 7r : B -t A and CY : A -t B be *-homomorphisms such that CY o 7r

is homotopic to idB· Define a *-homomorphism 'r/: B -t B EB 7r(B) by

rJ(b) =CY o 7r(b) EB 7r(b).

Evidently this *-homomorphism is homotopic to idB EB 7r (which is faithful).

Hence the proof will be complete, appealing to the previous proposition,

once we observe that rJ(B) is QD. But rJ(B) ~ 7r(B) C A and A is QD by

assumption. D

With this theorem in hand we can give some new examples of QD C* -

algebras.

Corollary 7.3.7. For any C*-algebra A, both the cone over A,

CA= Co(O, 1] @A,

and the suspension of A,

SA= Co(O, 1)@ A,

are QD.

Proof. Since SA <J CA, it suffices to show that CA is homotopic to zero.
This is familiar to the K-theory crowd, but let's recall the proof. Define a
family of *-homomorphisms CYt: Co(O, 1] -t Co(O, 1] by

CYt(f)(s) = f(ts),

for 0 :St :S 1. One checks that each CYt is a *-homomorphism and that this
family defines a homotopy between the zero map and the identity. Tensoring
with idA shows that CA is homotopic to zero. D

We will actually use the corollary above in the nonseparable setting, so
you may want to convince yourself that this case can be deduced from the
separable one.