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
® 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
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
~ 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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