434 17. K-HomologyProof. Let 'ljJ: A --+ Q(1i) be a unital *-monomorphism and O": A --+ JIB(JC)
be a unital faithful essential representation. Let f; c JIB(H) be the pullback
of 7/J(A) c Q(1i) and consider the representation
O" 0 7/J-l 0 1r1-(,: f;--+ JIB(JC).
It suffices to show that the natural inclusion i: & '---+ JIB(H) is approximately
unitarily equivalent relative to the compacts to the map
i EB ( O" o 'ljJ-^1 o nH): & --+ JIB(H EB JC).
This, however, is immediate from Theorem 1.7.3. DIt turns out that it is easier to characterize the invertible elements of
Ext(A) than it is to decide whether or not every element is invertible, so
let's tackle the easy question first.
Definition 17.1. 7. A unital *-monomorphism <p: A --+ Q(1i) is called
liftable if there exists a u.c.p. map O": A --+ JIB(H) such that 1rH o O" = <p.Proposition 17.1.8 (Invertible elements in Ext(A)). [<p] E Ext(A) is in-
vertible if and only if <p: A--+ Q(H) is liftable.Proof. We leave it to the reader to check that one representative of a Busby
invariant is liftable if and only if all other representatives have this property.
First suppose that [<p] E Ext(A) is invertible. Then there exists an
element [7/J] E Ext(A) such that [<pEB7/J] = 0 E Ext(A). In other words, there
exists a unital faithful essential representation 'fJ: A--+ JIB(H EB JC) such that
1rHffiJC o 'TJ(a) = <p(a) EB 7/J(a) E Q(1i EB JC).
Let PH E JIB(H EB JC) be the projection onto 1-{ EB 0. One checks that our
desired u.c.p. map O": A--+ JIB(H) is given by O"(a) = P'f}(a)P.
For the converse, assume that <p: A --+ Q(1i) is liftable and let O": A --+
JIB(H) be a u.c.p. lifting. Let rJ: A --+ JIB(JC) be the Stinespring dilation
of O" and P E JIB(JC) be the Stinespring projection (so we identify O" with
a 1-> Pry(a)P). Define a u.c.p. map 1: A--+ JIB(P..LJC) by 1(a) = p..Lry(a)P..L.
The key observation is that I is a -homomorphism modulo the compacts
(i.e., 1(ab) - 'Y(a)r(b) is compact for all a, b EA). To see this, observe that
the -homomorphism <p can be identified with a 1-> nJC(P)nJC(ry(a))nJC(P) E
Q(JC). Thus, the calculation used to identify multiplicative domains of u.c.p.
maps shows that nJC(P) commutes with nJC(ry(A)). Hence [P, 'TJ(a)] E IK(JC),
for all a EA, and this implies/ is a -homomorphism modulo the compacts.
If/ happens to be a faithful -homomorphism modulo the compacts,
then we are done, and if not we simply add on a faithful essential represen-
tation to complete the proof. D