1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

320 11. Simple C* -Algebras

and c.c.p. maps am: A-+ Ms(m)(C), f3m: Ms(m)(C) -+A with all of the

following properties:

(1) llam(ab) - am(a)am(b)ll < 2 ;,. for all a,b E Sm with llall, llbll:::; 1;

(2) llf3m o am(a) - aJI < 2 ;,. and llf3m o am( ab) - abll < 2 ;,,, for all

a, b E Sm with llall, llbll :::; 1;

(3) Sm+l contains both of the following sets:

{ab: a,b E Sm} and {f3m(am(a))f3m(am(b)): a,b E Sm};

(4) the union of the Sm's is dense in A.

Just as in the proof of Lemma 7.5.5, the first two conditions ensure that

f3m is almost multiplicative on the unit ball of am(Sm)· More precisely, for

contractions a, b E Sm we have

The generalized inductive system we are after is given by the matrix

algebras Ms(m) (C) and connecting maps defined by

and

'Pn,m = 'Pn,n-1 ° · · ·^0 'Pm+l,m

for all n > m. Since f3m is almost multiplicative on am(Sm), our system

is asymptotically multiplicative and, hence, an honest generalized inductive

system. Indeed, also using the almost-multiplicativity of the am's, one finds

that ll'Pn,m('Pm,k(x)cpm,k(Y)) - 'Pn,k(x)'Pn,k(Y)ll is bounded above by

1 1

2m-2 + 2m+l + ll'Pn,m+i('Pm+l,k(x)cpm+l,k(Y)) - 'Pn,k(X)'Pn,k(Y)ll·

Repeating this inequality until we reach n, asymptotic multiplicativity is

established.

Showing that A is isomorphic to the resulting generalized inductive limit

C*-algebra is similar to Theorem 11.1.5 and will be left to the reader. D

Our next goal is· a local characterization of strong NF algebras, but we
first need a nontrivial perturbation lemma.
Lemma 11.2.6. For any finite-dimensional C-algebra A and c: > O, there
exists J = J (A, c:) > 0 such that for every u. c. p. map cp from A into a
finite-dimensional C -algebra B with

llidA 0 'P-^111 < 1 + J,

there exists a complete order embedding 'ljJ: A-+ B such that 111/J - 'Pll < c:.