1 Approximately absorbing homomorphisms

In this section, we introduce approximately absorbing homomorphisms and

give some of their elementary properties. We have not proved that homomor-

phisms from

C(S1)

® Om to a purely infinite simple C*-algebra B are auto-

matically approximately absorbing, so this condition appears as a hypothesis

in our most general theorems. However, we will see in this section that the

homomorphisms in the direct systems corresponding to the most interest-

ing cases (tensor products of even Cuntz algebras with irrational rotation

algebras, Bunce-Deddens algebras, etc.) are automatically approximately

absorbing.

R0rdam's work ([Rrl] and [Rr2]) is already needed to prove that a ho-

momorphism from Om to B is approximately absorbing. We will therefore

need to assume throughout this section that our Cuntz algebras are even.

We begin by establishing terminology and notation for approximate uni-

tary equivalence.

1.1 Definition. Let A and B be C*-algebras, let G be a set of generators

of A, and let p and ip be two homomorphisms from A to B. We say that p

and if) are approximately unitarily equivalent to within e, with respect to G,

if there is a unitary v £ B such that

\Mg) - vxf{g)v*\\ e

for all g G G. We abbreviate this as

p

~ if).

(Note that we have suppressed G in the notation.) We say that ip and \f) are

approximately unitarily equivalent if p ~ ift for all e 0. (Of course, this

notion does not depend on the choice of G.)

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