7.3. Homotopy invariance 249Proof. Let CC JIB(JC) be a faithful, essential representation and let O"t: B----+
C be a homotopy of -homomorphisms between O"Q and 0"1. It suffices to
show that for each finite set J c B of the unit ball and c; > 0 there is a
-homomorphism 1f: B----+ lffi(?-i) and a finite-rank projection P E lffi(?-i) such
that
\[P,7r(b)]\ Sc:
and
\P1f(b)P\ ~ \bl\ - c:,
for all b E J.
For any 6 > 0 (to be specified later) we can, by norm continuity of the
homotopy and compactness of [O, 1], find a large n with the property that
\O"i_(b) n - O"j+l n (b)\ s 6
for all b E J and 0 S j S n-1. Since O"o is injective, we can find a finite-rank
projection Q E JIB(JC) such that
\\QO"o(b)Q\\ ~ \\bl\ - sfor all b E J. Applying the existence of quasicentral approximate units to
the ideal of compact operators, we can find positive, norm one, finite-rank
operators
Q S Fo S F1 S F2 S · · · S Fn S 1such that Fj+lFj = Fj and
for all 0 s j s n and b E J. (Since any approximate unit of finite-rank
operators will eventually almost dominate Q, we can perturb a little to
actually dominate Q.)
Here is a crucial remark: Since 0"1 (B) is QD and C C JIB(JC) is essential,
we may assume that Fn is a projection! Indeed, the representation theorem
says we can find a finite-rank projection which almost commutes with 0"1 (J)
and is almost the identity on a prescribed finite set of vectors (e.g., a basis
for the range of whatever finite-rank operator Fn we are originally given).
The proof of Proposition 7.2.3 allows us to perturb and actually dominate
Fn and then replace Fn by such a projection.
Now comes the crucial trick: Consider the operator V: JC ----+ EB~ JC
defined by
Vh -o - c^1!^2 h rT\ w1 c^1!^2 h rT\ w ••• Wn' rT\ c^112 h
where Go = Fo, G1 = F1 - Fo, ... , Gn = Fn - Fn-1· A calculation shows
that
V*V = Fn E lffi(JC)