- C* -algebras associated to discrete groups 43
Example 2.5.1. If r = Z, then C~(r) = C('JI'), the continuous func-
tions on the circle. Indeed, the Fourier transform identifies .€^2 (Z) with
£^2 (11', Lebesgue) and one checks that this unitary takes C~ (Z) to ( contin-
uous) multiplication operators. More generally, for every abelian group r,
Pontryagin duality gives an identification of C~ (r) with C (f'), the continu-
ous functions on the dual group.The full (or universal) group C*-algebra of r, denoted C*(r), is the
completion of qr] with respect to the norm
llxllu =sup 7[ [17r(x)ll,where the supremum is taken over all (cyclic) -representations 7r: qr]--+
JIB(H). (Note that, since unitaries have norm one, the supremum is finite.)
Evidently C (r) enjoys the following universal property.
Proposition 2.5.2. Let u: r --+ JIB(H) be any unitary representation of
r. Then, there is a unique -homomorphism Ku: C(r) --+ JE(H) such that
1fu(s) =Us for all SE r.
In particular, notice that_ C(r) always has a character - i.e., a one-
dimensional representation - coming from the trivial representation r --+
C = JIB(C). (This also defines a tracial state on C(r) such that s 1-+ 1 for
all s E r.) The reduced group C-algebra C~(r) c JIB(.€^2 (r)) always has
a faithful trace (which is more important than the trivial trace on C (r)
described above).
Proposition 2.5.3. The vector state x 1-+ (xSe, Se) defines a faithful tracial
state on c~ (r).
Proof. A simple calculation shows this state to be tracial.
Clearly Ps commutes with every operator in C~(r) (since this is easily
seen on the generators A. 9 E C~ (r)). It follows that Se is a separating vector,
meaning that xSe = ySe if and only if x = y (for all x, y E C~(r)). Indeed,
if xSe = ySe, then
xSs = p:xse = P:Yse = ySs,
for alls Er. Since such vectors span .€^2 (r), it follows that x = y. With this
observation, faithfulness is simple: If 0 ::; x E C~ (r) and 0 = (xSe, Se), then
0 = llx^112 Sell and this implies x^112 = 0. Thus x = 0 too. D
The group von Neumann algebra of r is defined to be
L(r) = c~(r)" c JIB(.e^2 (r)).Though we won't give the proof until Chapter 6, a fundamental theorem
of Murray and von Neumann states that L(r) is the commutant of the