460 C. Lifting Theorems
To prove the claim, we proceed by induction. First, set 1/;1 = 1/Ji. Sup-
pose now that we have constructed 1/;1, ... , 1/Jn with the desired property.
Let (e>,)A be a quasicentral approximate unit of Jin B. Then, fork:::;; n,
lim 11(1-e>,)^112 1/Jn(xk)(l - e>,)^112 + e~^12 1/Jn(xk)e~^12 -1/Jn(xk)ll = 0
.\and for bk = 1/J~+l (xk) -1/Jn(xk) with k :S: n,
lim 11(1-e>,)^112 bk(l - e.>,)^11211 = Jl7r(bk)ll < 3/2n+l.
.\
Hence, we can take e = e .\ E J so that for every k :::;; n
11(1-e)^112 1/Jn(xk)(l - e)^1 /^2 + e^112 1/Jn(Xk)e^112 -1/Jn(xk)ll < 1/2n+land
11(1 - e)^112 bk(l - e)^1 /^2 ll < 3/2n+l.
Then the c.c.p. map 1/Jn+i: E--+ B defined by
1/Jn+i(x) = (1-e)l/21/;~+l(x)(l - e)l/2 + el/21/Jn(x)el/2
satisfies the desired property. (Note that 7r o 1/Jn+i = 7r o 1/J~+i ·) D
We observe that if .a c.c.p. map cp: E --+ B / J has a c.p. lifting, then
it actually has a c.c.p. lifting. Indeed, if E is unital and 1/J: E --+ B is a
c.p. lifting of cp, then for an approximate unit (f >-h in J, the c.c.p. maps
1/J>-( ·) = (l-e),)'lj;( · )(1-e.>,) are c.p. liftings of cp with lim.>, 111/J>-ll = 1. When
E is a nonunital C -algebra, the proof is more cumbersome and we leave it
to the reader.
The following result is due to Choi and Effros.
Theorem C.3. Every nuclear c.c.p. map from a separable C-algebra A
into a quotient C -algebra. B / J is liftable. In particular, every c. c. p. map
from a separable nuclear C -algebra is liftable.
Proof. By Lemma C.2, the set of liftable c.c.p. maps is closed in the point-
norm topology. Since a nuclear c.c.p. map is approximated by c.c.p. maps
which factor through full matrix algebras, it suffices to show that every
c.c.p. map cp from a full matrix algebra Mn(C) into B / J is liftable. We
use Proposition 1.5.12. Let a= [cp(ei,j)] be a positive element in Mn(B/J).
Since 7rn: Mn(B)--+ Mn(B/J) is a surjective *-homomorphism, the positive
element a lifts to a positive element b = [bi,j] E Mn(.8) and the corresponding
map 1/J': Mn --+ B is a c.p. lifting. D
The separability assumption in Theorem C.3 is essential. .Indeed, there
is no bounded linear lifting from .€^00 /co to .€^00 (see Exercise 13.1.1).
Here is a celebrated lifting theorem due to Effros and Haagerup.