136 4. Constructionsalgebras. Can you find the finite graph which yields 02? Hint: It has a
single vertex!)
Though the details require plenty of calculations, here is a description
of the fixed point subalgebra of the gauge action on C*(lB). Consider the
norm closure of the set
span{(se 1 Se 2 • • • Sen)(s1is12 · · · SJn)*: ei, fj EE, 1 :S i,j :Sn EN}.
Thanks to the relations in the definition of C ( 6), it can be shown that
this is actually a C-subalgebra (similar to the case of Cuntz algebras); it
is clearly contained in the fixed point subalgebra of the gauge action; and,
using the conditional expectation, it is not too hard to show that this is the
entire fixed point subalgebra (since the conditional expectation maps words
into this set). One then must prove that we are actually looking at an AF
algebra which, though slightly more technical, is also similar to the case of
Cuntz algebras. See Corollary 3.3, and the discussion after it, in [163] for
more details.
If you are willing to believe this, then we have the following corollary.
Corollary 4.5.4. For any row finite graph 6, there is a canonical gauge
action 11' -+ Aut ( C* ( 6)) whose fixed point subalgebra is AF. It follows that
C*(lB) is nuclear.In [163, Remark 4.3], an alternate proof for nuclearity is given. It turns
out that JK@C(lB) ~ A><1aZ for an AF algebra A (in fact, A= C(lB) ><l'Y 11').
This proof has the added benefit of showing that graph algebras always sat-
isfy the Universal Coefficient Theorem of Rosenberg and Schochet ([170]).
4.6. Cuntz-Pimsner algebras
We now wish to study nuclearity and exactness in the context of a very
general construction due to Mihai Pimsner. Most of this section is spent
describing the construction and proving requisite facts. The main theorem
for us - both nuclearity and exactness are preserved by this construction:
Theorem 4.6.25 - comes at the very end. Unfortunately, tons of technical
results are needed to get there. In order to keep this section less than 1,000
pages, we will be a bit terse. For those primarily interested in groups and
their actions, there is little harm in jumping to Chapter 5 and referring back
as necessary.
Preliminaries on Hilbert C*-modules. Pimsner's construction requires
an understanding of the basic theory of Hilbert modules. We state the facts
we need; see [114] for proofs.