42 2. Nuclear and Exact C*-Algebras2.5. C*-algebras associated to discrete groups
This section contains a bare-bones introduction to an important class of
C -algebras. There is nothing about nuclearity or exactness until the next
section, to which you should proceed if already familiar with the basics of
group C -algebras.
For a discrete group r we let A.: r -t JIB(£^2 (r)) denote the left regular
representation: A.s(5t) = Ost for alls, t E r, where {5t : t E r} C £^2 (r) is
the canonical orthonormal basis. There is also a right regular representation
p: r -t JIB(£^2 (r)), defined by Ps(5t) = 8ts-i. Note that A. and pare unitarily
equivalent; the intertwining unitary is defined by U Ot = Ot-1.
We denote the group ring of r by qr]. By definition, it is the set of
formal sums
Lass,
sEr
where only finitely many of the scalar coefficients as E CC are nonzero, and
multiplication is defined by
(L ass)(L att) = L asatst.
sEr tEr s,tEr
The group ring qr] acquires an involution by declaring (l::sEr ass r =
l::sEr ass-^1. Note that the left regular representation can be extended to
an injective -homomorphism qr] __, JIB(£^2 (r)), which we also denote by A..
Evidently, there is a one-to-one correspondence between unitary representa-
tions of r and -representations of qr].
Both amenable and exact groups are defined in terms of their canonical
actions on £^00 (r). For f E £^00 (r) and s E r we let s.f E R^00 (r) be the
function s.f(t) = f(s-^1 t); simple calculations show that f 1-+ s.f defines
a group action of r on £^00 (r). An important fact is that this action is
spatially implemented by the left regular representation. That is, if we regard
£^00 (r) c JIB(£^2 (r)) as multiplication operators (i.e., f5t = f(t)8t), then a
calculation shows
for all f E f^00 (r) and s Er.
The reduced C* -algebra of r, denoted C~ (r),^11 is the completion of qr]
with respect to the norm
llx/lr = llA.(x)llllll(£2(r))-
Though isomorphic to C~ (r), it is sometimes useful to consider c; (r), which
is just the closure of qr] in the right regular representation.
(^11) You will also see c;(r) in the literature.