1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.2. Cones over exact RFD algebras 267

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8.2. Cones over exact RFD algebras

A few more fairly simple facts and we'll be able to AF embed the cone

over an exact RFD algebra. The first step is to observe the existence of

(i, c )-admissible representations.

Proposition 8.2.1. Let A be exact and residually finite-dimensional. For

each finite set i CA and c > 0 there exists an (i, c)-admissible representa-

tion CT : A ---+ Mk ( C).

Proof. This is an immediate consequence of Corollary 7.5.8 since there is

only one way to put a matrix algebra (unitally) inside of IIB('H). D

We now list, without proof, two trivial facts which are important below.

Lemma 8.2.2. Let CT: A ---+ Mk(C) be an (i, c)-admissible representation

and 7r: A---+ MN(C) be any other representation. Then CT EB 7r is also (i, c)-

admissible.

Lemma 8.2.3. Let CA denote the unitization of the cone over some C*-

algebra A. Any two unital *-representations of CA on the same Hilbert space

are homotopic.

Theorem 8.2.4 (Dadarlat). Let A be a separable exact residually finite-

dimensional C* -algebra. Then the cone over A is AF embeddable.^1

Proof. The proof amounts to reviewing everything we have done so far.

Let CA be the unitization of the cone over A and in C CA be an in-

creasing sequence of finite subsets whose union has dense linear span. Let

En= 2 1;. and let Pn: CA---+ Mk(n)(C) be a sequence of (in,cn)-admissible

representations. Note that the Pn's are automatically asymptotically iso-

metric.

We now mimic the proof of Proposition 8.1.3 to construct a sequence

of finite-dimensional C*-algebras Bi, injective *-homomorphisms 7ri: Bi ---+

Bi+l and *-homomorphisms CTi: CA---+ Bi such that each CTi contains Pi as

a direct summand and

00

L llCTi+i(a) -1fi o CTi(a)ll < oo

i=l

for every a E LJ in· As before, this will imply AF embeddability.

lone can even embed it into the UHF algebra of type 200 , but we won't prove this. (Our

construction only gives a UHF embedding.) It requires a bit more work, keeping in mind that

the unitized cone has a one-dimensional representation that can be used to modify dimension -

i.e., increase the dimension of a given representation by taking direct sums with one-dimensional

representations.