1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

270 8. AF Embeddability

suggests the possibility of adapting Proposition 8.1.3 to deduce AF embed-
dability of all such algebras. You may want to figure out where the proof
breaks down; it isn't necessary, but it will help motivate our next step for
coned algebras.

For a unital separable exact C* -algebra A and t E [O, 1] we let

C(t) = {f E C([t, 1], A) : f(l) E ClA}·

Note that C := C(O) is just the unitization of the cone CA over A and
C(l) = C.^2 We denote by 1C(t): C --+ C(t) the natural restriction homo-
morphism from C onto C(t) and let Cf(t): C(t)--+ C be the *-homomorphism
given by

(ii(t)(f))(s) = {f(t) ~f s < t,

f(s) ifs2:t

for f E C. Clearly O"(t) := if(t)1C(t) is a continuous path of endomorphisms
on C with O"(O) = idc and O"(l)(C) = Cle. We will need to smear some
functions around, so if i c C and P c [O, 1] are given finite sets, we let

ip = {O"(p)(f): f E i, p E P} c C.

Note that O"(p)(i'P) c iP for every p E P.

The next lemma is the key technical fact. The proof resembles your
worst nightmare, but the main point is this: Uniqueness of unital maps
from C to a matrix algebra is automatic! Very roughly, we'll slowly work
our way down the cone, applying stable uniqueness over and over, until we
arrive at (['. where automatic uniqueness saves the day.

Lemma 8.3.4. Let C be as above and suppose we are given

(1) a finite set io of unitaries in C;

(2) co > O;

(3) afinite setP = {po,P1, ... ,PN} c [O, 1] such that pi:::; Pi+b Po= 0,

PN = 1 and llf(s) - f(t)ll <co for all f E io and Pi S s, t:::; Pi+l,

i = 1, ... ,N -1.

Then there is a 5o > 0 with the following property: For every

• collection of (1C(Pi)(t6), 5o)-suitable u.c.p. maps Bo(i): C(pi) --+
  Do(i), where Do(i) is a matrix algebra and i = 1, ... , N,


• finite set of unitaries ii C C and


• 51 > 0,

(^2) We have reversed things and unitized C([O, 1),A).