1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.5. C*-algebras associated to discrete groups 45

Definition 2.5.6. A function <p: r ____, C is said to be positive definite if the

matrix

[cp(s-^1 t)]s,tEF E Mp(C)

is positive for every finite set F c r.

Fix a positive definite function <p and let Cc(r) be the finitely supported

functions on r. Define a sesquilinear form Cc(r) x Cc(r) ____, C by

U, g)cp = L <p(s-^1 t)f(t)g(s).

s,tEr

This form is positive semidefinite. Indeed, if f E Cc(r) has support F, then

(!, f)cp = L cp(s-^1 t)f(t)f(s) = ([cp(s-^1 t)]s,tEF(f), (f) ),

s,tEr

where the inner product on the right is the standard one on £^2 (F). Since <p
is positive definite, (!, f)cp :2: 0 as asserted. Hence we can mod out by the
zero elements and complete to get a Hilbert space £~(r). For f E Cc(r) we

let J E £~(r) denote its natural image. Here's a GNS construction for the
present context.

Definition 2.5.7. If <pis a positive definite function on r, then )..'P: r ----t
IIB(£~(r)) is the unitary representation given by >..f (f) = :;J, where s.f(t) =
f(s-^1 t), for all t E I'.^14

Note that

(>..f Je, Je)cp = (Js, Je)cp = cp(s),

for alls Er, and hence we recover <p from the vector functional (· 8e, 8e)·

Perhaps the construction of £~(r) seems familiar? It should. Suppose <p
is a positive linear functional on C*(I'). Thens 1---+ cp(s) is a positive definite
function on I': for s1, ... , Sn E I' we have

which is positive since <p is a c.p. map. It is a simple exercise to show that
the GNS space of C*(I') with respect to <pis nothing but £~(r).

Why then have we introduced positive definite .functions? It is often
necessary to work only with r, instead of some bloated C* -algebra. For
example, here is a useful application.

14It isn't hard, just tedious, to check that this is really a unitary representation.