1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

376 13. WEP and LLP

A unital C*-algebra A has the lifting property (LP) (resp. local lifting prop-

erty (LLP)) if any c.c.p. map from A into a quotient C*-algebra B/J is

liftable (resp. locally liftable). A nonunital C*-algebra A has the LP (resp.

LLP) if its unitization has that property.

Not surprisingly, we can usually restrict our attention to u.c.p. maps.

Lemma 13.1.2. A unital C*-algebraA has the LLP (resp. LP) if any u.c.p.

map from A into a quotient C* -algebra B / J is locally liftable (resp. liftable

and A is separable).

Proof. Suppose that a c.c.p. map <p: A -----+ B / J is given and let .c =

1.p(l) E B/J. Let Cn = hn(c) for the positive continuous function hn(t) =

max{ t, 1/n }. Fix a state w on A and define u.c.p. maps !.pn: A -----+ B / J by

l.f)n(a) = c;:;-^112 (<p(a) +w(a)(cn - c))c;:;-^112.

By assumption, l./)n has a (local) lifting 'I/Jn into B. Let bn E B be any lifting

of Cn with 0 ::;: bn ::;: 1. Then, one has

7r(b;/^2 '1/Jn(a)b;/^2 ) = 1.p(a) + w(a)(cn - c)-----+ 1.p(a)

for every a EA. Hence, by Lemma C.2, <pis (locally) liftable. D

The Choi-Effros Lifting Theorem (Theorem C.3) implies that separable

nuclear C* -algebras have the LP. Here is a nonnuclear example:

Theorem 13.1.3 (Kirchberg). The full C*-algebra C*(lFn) of a countable

free group lFn has the LP.

Proof. Assume n = oo (the finite case is identical). We first show that a*-

homomorphism e : C* (JF n) -----+ BI J is liftable. To this end, let XI' X2' ... E B

be contractive liftings of B(U1), B(U2), ... , where U1, U2, ... are the standard

generators of C*(lFn)· Then, each Xn dilates to a unitary

A _ [ Xn (1 - XnX~)^112 ]

Xn - (l _ XnXn * )1/2 _ Xn * E M2(B).

By universality, there is a unital -homomorphism p: C(lFn)-----+ M2(B) with
p(Un) = Xn. It is not hard to see that the (1, 1)-corner of pis a u.c.p. lifting
of e, so the homomorphism case is complete.
Now, let <p: C(lFn) -----+ B/J be a u.c.p. map. Since lFn is countable,
we may assume B / J is separable. By the Kasparov-Stinespring Dilation
Theorem [97], there is a -homomorphism e: C(lFn)-----+ MCIK.@(B/J)) such
that 1.p(a) = B(a)u for a E C(lFn), where xu is the (1, 1)-entry of x in the
multiplier algebra of OC ® ( B / J). By the noncommutative Tietze extension
theorem (Proposition 6.8 in [114]), the surjective -homomorphism id® 7r
from OC ® B onto OC ® (B / J) extends to a surjective -homomorphism if-