262 8. AF Embeddability
Proposition 8.1.1 (Approximately commuting diagrams). Given a C-
algebra A, assume there exist finite-dimensional C -algebras Bi, injective
*-homomorphisms 'Tri: Bi --+ Bi+l and u.c.p. maps O'i: A--+ Bi such that
(1) the O'i 's are asymptotically isometric and asymptotically multiplica-
tive;
(2) there exists a set 6 C A with dense linear span, such that for all
a E 6,
00
L ll7ri(O'i(a)) - O'i+i(a)ll < oo.
i=i
Then A is AF embeddable.Proof. Here's a sketch. First, let B be the inductive limit of the sequence
Bi~B2~B3~···.
Since the maps 'Tri are assumed injective, we may (and will) identify each Bi
with a subalgebra of B. Then we can regard each map O'i as taking values
in Band one uses hypothesis (2) to show that the sequence {O'i(a)} CB is
Cauchy for every a E 6.
Letting X c A be the linear span of 6 (which is assumed dense in A),
it follows that { O'i ( x)} C B is Cauchy for every x E X. Thus we can define
(J': X--+ B by
<J'(x) = _lim O'i(x).
i-+oo
Now one checks that <J' is an isometric linear map; hence it has a unique
(isometric) extension to all of A. Finally one checks that this extension
has no choice but to be a -homomorphism, thereby giving a C -algebraic
embedding of A into B. (There are a number of things to check, but they're
~~y) D
Sometimes variations of the previous result get used in providing AF
embeddings. For example in the Pimsner-Voiculescu AF embedding of irra-
tional rotation algebras there are no maps O'i. However, the core idea is the
same as they construct matrix models which converge in norm and asymp-
totically satisfy the defining relations of an irrational rotation algebra; the
embedding then comes from universality. However, we'll use the previous re-
sult explicitly; hence we'll need an appropriate "uniqueness" result in order
to invoke it. The following notation will be handy and used repeatedly.
Definition 8.1.2. Given <J': A--+ Mk(q and a positive integer N,
NO': A - MNk(C)denotes the N-fold direct sum of <J'. We will often identify NO' with the map
<J' 0 lN: A--+ Mk(C) 0 MN(C).