1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

386 13. WEP and LLP

Lemma 13.4.3. For every x E J, a E F and f E Co(O, 1], we have

(1) limn-+oo llPn(f)xll = 0,

(2) PnU)a - fa E CJ for every n EN, and

(3) limn-+oo llPn(f)a - apn(f) II = 0.

Proof. These assertions are all trivial when f is a polynomial with f (0) = 0.

Since such polynomials are dense in Co ( 0, 1], we are done. D

For a C*-algebra D, we set Dao = rr~=l D/ EB~=l D and define a *-

homomorphism Poo by

00

Poo: Co(O, 1] 3 J f-+ (pn(f))n +EB CF E (CF) 00 •

n=l

We regard F C (C[O, 1] @ F) 00 as constant functions. Since the range of

p 00 commutes with F, part (3) of Lemma 13.4.3 implies that they give rise

to a *-homomorphism <p = p 00 x idp: CF-+ (CF) 00 • Since the C*-algebra

CF has the LP, <p has a c.c.p. lifting 'ljJ: CF -+ IT~=l CF. It is not hard

to see that ker <p = CJ and hence <p induces an injective *-homomorphism

CA'--+ (CF) 00 • Indeed, we have the following stronger result.

Lemma 13.4.4. For any C*-algebra D, the c.c.p. map 'l/;@idn induces an

isometric *-homomorphism ():

Proof. For any f E Co(O, 1] and a E F, we have 'ljJ(fa) = (pn(f)a)n modulo

EB~=l CF. Hence, 'ljJ is multiplicative modulo EB~=l CF and 'ljJ maps CJ

into EB~=l CF by part (1) of Lemma 13.4.3. It follows that () is a well-

defined *-homomorphism. Moreover, for any y = L:k fkak@ Xk E CF@D,

we have

ll()(Q(y))ll = limsup II LPn(fk)ak @xkll 2". llQ(y)JI

n-+oo k

by part (2) of Lemma 13.4.3. This implies that () is isometric. D

Lemma 13.4.5. For any separable D, there is a c.c.p. map '1!: CF -+
rr~=l Mn(<C) such that [f is multiplicative modulo EB~=l Mn(<C), [f maps