1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.3. Cones over general exact algebras 271

there is an (.~1, 51)-suitable u.c.p. map 81 from C into a full matrix algebra
D1 and positive integers no(i) EN (i = 1, ... , N) such that

(-~'/~',4c:o) f£

81 Rj wno(i)eo(i),

i=l

where 8o(i) = eo(i) o n(pi) for every i.

Proof. Let h( s) = 5y's and hj = h o · · · o h be the composition of h with
itself j times. We can take any 5o such that hN(5o) < Eo.

Let the eo(i)'s, ~1 and 51 be given. We may assume that ~1 ==:> t6 and
51 < 5o. Since each C(pi) is exact and QD, we can find (n(pi)(~1), 51)-
suitable maps D(i) from C(pi) to matrix algebras, for i = 0, 1, ... , N - 1.
Let v(i) = D(i) on(pi) and note that v(O) is an (~1, 51)-suitable map. Hence
it will suffice to construct 81 of the form

81 = v(O) EB m(l)v(l) EB··· EB m(N - l)v(N - 1),

for some positive numbers m(l), ... , m(N -1).

Applying Lemma 8.3.3 to the (n(p1)(~l;'), 5o)-suitable map eo(l) and the
(n(p1)(~6), 5o)-multiplicative map v(O)oO-(p1), we find an integer no(l) such
that
no(l)eo(l) (7r(pi)(~),h(oo)) v(O) o O-(p1) EB ,0(1),
for some (n(p1)(~6), h(5o))-multiplicative map p(l) from C(p1) to a matrix
algebra. Evidently this implies
ctg,h(oo))
no(1)8o(l) Rj v(O) o O"(p1) EB p(l),

where p(l) = p(l) o n(p1).

Now one should apply Lemma 8.3.3 to the (n(p1)(~6), 5o)-suitable map

D(l) and the (n(p1)(~6), h(5o))-multiplicative map p(l) to find m(l) such

that

m(l)D(l) (7r(pi)(~iJ,h

2

(oo)) p(l) EB μ(1),

for some (n(p 1 )(~6), h^2 (5o))-multiplicative map μ(l). Thus

(:;16 ,h^2 (8o))

m(l)v(l) Rj p(l) EB μ(1),

where μ(1) = μ(1) o n(p1).

Let's take a moment to absorb what we've done. Using the fact that

(~f;,c:o)

v(O) Rj v(O) o O"(p1),

we see that

v(O) EB m(l)v(l) (~b Rj ,c:o) v(O) o O"(p1) EB ( p(l) EB μ(1) )