1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

13.1. The local lifting property 377

between their multiplier algebras. Hence, by the first part of this proof,

there is a u.c.p. map p: C*(lFn)--+ M(IK®B) such that()= if op. The u.c.p.

map 'f: C*(lFn) 3 a 1-+ p(a)11 EB is our desired lifting of <p. D

Drawing a diagram, one easily deduces the following fact.

Corollary 13.1.4. Let A be a separable C* -algebra and J be a closed two-

sided ideal in C*(JF 00 ) such that A~ C*(JF 00 )/ J. Then, A has the LP (resp.

the LLP) if and only ifidA: A--+ C*(JF 00 )/ J is liftable (resp. locally liftable).

Remark 13.1.5 (LP versus LLP). The full C*-algebra of an uncountable

free group has the LLP (but probably does not have the LP). Indeed, when

lF1 is the free group on an uncountable set I, every element x (resp. every

separable C*-subalgebra A) in C*(lF1) sits inside C*(JFI') for some countable

subset I' C I. Moreover, C* (JF I') C C* (JF I) is the range of a conditional

expectation. Hence the LLP of C* (JF I) follows from that of C* (JF I').

There is a (nonseparable) C*-algebra which has the LLP but not the

LP. Indeed, the commutative C* -algebra goo/ co is such an example as there

is no bounded linear lifting from goo /co into goo (Exercise 13.1.1). On the

other hand, an affirmative answer to the QWEP conjecture (see Section

13.3) would imply that the LP and the LLP are equivalent for separable

C*-algebras (cf. [102]). ·

Recall that in the particular case of quotients, Effros and Haagerup gave
a tensorial characterization of local liftability (Theorem C.4):

Theorem 13.1.6. Let J be a closed two-sided ideal in a unital C* -algebra B

and let 7r: B --+ B / J be the quotient map. Then, the following are equivalent:

(1) for any C*-algebra A, the following sequence is exact:

o---A® 1---A0B---A® (B/J)---0;

(2) same as above but with A= JE(g^2 );

(3) the identity map on B / J is locally liftable.

Note that condition (1) holds whenever A ®max (B/J) =A® (B/J)
(Corollary 3.7.3). Thus, if B/J 0 JE(g^2 ) has a unique C-norm, it follows
that id BI J is locally liftable; Corollary 13.1.4 then implies that B / J has the
LLP (since we may assume B = C(JF 00 )). In the next section we prove the
converse.

Exercise

Exercise 13.1.1. Prove that co(IR) (where IR is just viewed as an uncount-
able set) is -isomorphic to a C-subalgebra of goo /co. Conclude that there
is no injective continuous linear map from goo/ co into goo.