1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

272 8. AF Embeddability

= (v(O) o o-(p1) EB p(l)) EB μ(1)

(tg ,h(8o))

~ no(l)Oo(l) EB μ(1).

Note that we've taken a step to the right in the sense that the "remainder"

term μ(1) factors through C(p1).

Another application of Lemma 8.3.3 to the (7r(p2)CJ"{{'), 60)-suitable map

Bo(2) and the (7r(p2)(~6), h^2 (<5o))-multiplicative map μ(1) o O-(p2) yields an

integer no(2) such that

(';f6,h^3 (8o))

no(2)0o(2) ~ μ(l) o o-(p2) EB p(2),

where p(2) = j5(2) o 7r(p2) for some (7r(p2)(~6), h^3 (<5o))-multiplicative map

j5(2). It follows that

no(l)Oo(l)EBno(2)0o(2)

(tg,~(oo)) (v(O) o o-(p 1 ) EB p(l)) EB (μ(1) o a(p2) EB p(2))

= v(O) o O"(p1) EB (p(l) EB μ(1) o a(p2)) EB p(2)

Ct6,eo) ~ v(O) EB ( p(l) EB μ(1) ) EB p(2)

(t6,h^2 (8o))

~ v(O) EB m(l)v(l) EB p(2).

Combining these estimates, we have

(;y'b ,3eo)

v(O) EB m(l)v(l) EB p(2) ~ no(l)Oo(l) EB no(2)0o(2),

where the term p(2) factors through C(p2); thus we've moved one step fur-

ther to the right.

Now we repeat this procedure ad nauseam: Absorb p(2) into a multiple

of v(2) with a remainder μ(2), which factors through C(p 2 ), and then check

that

v(O) EB m(l)v(l) EB m(2)v(2) (;y'b ~ ,3eo) ( ~ ~ no(i)Oo(i) ) EB μ(2).

Then absorb μ(2) o O"(p3) into a multiple of Oo(3) and find that

(%6',3eo) ~

v(O) EB m(l)v(l) EB m(2)v(2) EB p(3) ~ W no( i)Oo( i),

i=l

with a remainder p(3) which now factors through C(p 3 ), and so on.