212 6. Amenable TracesA 2
Let T be a tracial state on A, L^2 (A, T) be the GNS space, AC L (A, T)
be the canonical image of A and n 7 : A -----t IIB(L^2 (A,T)) be the GNS repre-
sentation. We will often refer to n 7 as the left regular representation.
One defines a *-representation n~P: A^0 P -----t IIB(L^2 (A, T)) by
n~P(a)b = ba
for all a EA and b EA C L^2 (A, T).^1 We will refer to this representation of
A^0 P as the right regular representation.
Proposition 6.1.2. For a C* -algebra A and tracial state T the following
assertions hold:
(1) the left and right regular representations commute - i.e., n 7 (A)' :J
K~P(AoP);
(2) there is a unique conjugate linear isometry J: L^2 (A, T) -----t L^2 (A, T)
such that J(b) = C* for all b EA C L^2 (A, T) {also note that J^2 = l);
(3) for all a EA we have Jn 7 (a) = n~P(a*)J and Jn 7 (a)J = n~P(a*).
In particular, Jn 7 (A)J = n~P(A^0 P).Trivial calculations yield a proof. The next lemma is only slightly harder.
Lemma 6.1.3. For a C* -algebra A and tracial state T the following asser-
tions hold:
(1) for all vectors v,w E L^2 (A,T) we have (Jv,w) = (Jw,v);
(2) for arbitrary y E n 7 (A)' we have Jyi = y*i, where i E L^2 (A, T) is
the canonical cyclic vector (though we don't assume A is unital).Proof. The first assertion is easily verified on vectors of the form v = a and
w = b so a simple approximation argument handles the general case. The
second statement follows from the first and a little calculation:
(Jyi, b) = (Jb, yi)
= (y*nAb*)i, i)
= (nT(b*)y*i, i)
= (y*i, b).Since this holds for arbitrary b E A, we are finished. D
Since J^2 = 1, it is apparent that the map T r-+ JT J is a conjugate
linear *-isomorphism of IIB(L^2 (A, T)). In particular, this morphism preserves
commutants in the sense that for each set Sc IIB(L^2 (A, T)) we have JS' J =
(^1) It's worth checking that this really works! For example, even the fact that 7r~P(a) is well-
defined depends on T being a trace.
(^2) This means J(>-.v) =Xv for all).. E <C and v E L (^2) (A,T).