C. Lifting Theorems 461Theorem C.4. Let J be a closed two-sided ideal in a unital C* -algebra B
and 7r: B ---+ B / J be the quotient map. Then, the following are equivalent:
(1) for any C*-algebra A, the following sequence is exact:
0 --t A 0 J --t A 0 B --t A 0 (BI J) --t O;
(2) same as above but with A= JIB(£^2 );
(3) for any finite-dimensional operator system EC B / J, the inclusion
of E into B / J is liftable.Proof. Obviously, (1) implies (2). Let's prove (3) implies (1). Given z E
ker(idA 0 7r) c A 0 B, we must show z E A 0 J. For any E > 0, there
is y = :Ei ai 0 Xi E A 0 B with llY - zll < E. Let E C B / J be a finite-
dimensional operator system containing the 7r(xi)'s and let 'I/;: E---+ B be a
u.c.p. lifting. It follows that y - (idA 0 'I/; o 7r) (y) E A 0 J and
ll(idA07/J^0 7r)(y)ll '.S ll(idA07r)(y)ll '.S lly-zll <E.
Therefore,
dist(z,A 0 J) :S llz-(y-(idA 0 'I/; o 7r)(y))ll < 2c.
Since E > 0 was arbitrary, z EA 0 J as desired.
We now prove that (2) implies (3). Let E c B/J be given. By operator
space duality, the inclusion E C B / J corresponds to an element z E E 0
(B/J) with llzll = 1. We may assume E C JIB(£^2 ). Since
(E 0 B)/(E 0 J) c (JIB(£^2 ) 0 B)/(JIB(£^2 ) 0 J) = JIB(£2) 0 (B/J)
isometrically (Proposition 3.7.2), we have E 0 (B/ J) = (E 0 B)/(E 0 J)
isometrically. Hence, for any E > 0, one can lift z to an element z EE 0 B
with llzll < 1 + E. Then, the map 'I/;': E---+ B corresponding to z is a lifting
of <p with 117/J'llcb < 1 + E. We may assume that 'I/;' is self-adjoint. Since
'l/;'(1) -1 E J, we can find e E J with 0 :Se :S 1 such that
11(1-e)^112 ('1/;'(1) -1)(1-e)^1 /^2 ll < E.
Choose a unital positive linear functional e on E and let 'I/;": E ---+ B be
defined by
7/J"(x) = (1-e)^112 ( 7/J'(x) - O(x)('l/;'(1) -1)) (1-e)^112 + O(x)e
= [(1 - e)l/2 el/2] [7/J'(x) 0 ] [(1-e)l/2]
0 e(x) e^112
+ e(x)(l - e)^112 ('1/;'(1) -1)(1-e)^112.It follows that 'I/;" is a lifting of cp with 'l/;"(1) = 1 and 117/J"llcb < 1+2c. By
Corollary B.11, there exists a u.c.p. map 'I/;: E ---+ B such that 117/J -7/J"llcb <
4dim(E)c. Since E > 0 was arbitrary, we are done by Lemma C.2. D