1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

9.3. Equivalence of exactness and property C 295

representation v of JR. on lBl(H) -i.e., v(t)av(t)* =at( a) for every t E JR. and
a E M. We denote by .A the regular representation of JR. on L^2 (JR.), the
Hilbert space of square integrable functions on JR. with respect to Lebesgue
measure. The crossed product von Neumann algebra M ><la IR is (isomorphic
to) the von Neumann subalgebra in lBl(H 0 L^2 (JR.)) generated by (v 0 >.)(JR.)
and M 0 <Cl.

Lemma 9.3.6. There exists a conditional expectation E from M ><la JR. onto
M. Moreover, E is a point-ultraweak limit of normal u.c.p. maps from
M ><la JR. into M.

Proof. For each n, fix a unit vector fn E L^2 (JR.) with supp(fn) C [-1/n, l/n]
and define an isometry Vn: 1t ---t L^2 (JR., H) ~ 1t 0 L^2 (IR) by

for e E H. We define a normal u.c.p. map 'Pn: M ><la JR. ---t lBl(H) by 'Pn(x) =
v;xVn· It is routine to check that

(cpn((a 0 l)(v 0 .A)(s))11,e) = l fn(t)fn(t-s)(a-t(a)1J,e) dt

for every a E M and every s E JR.. In particular,

CfJn(a 0 1) = l lfn(t)l^2 a-t(a) dt

for a E M. Hence, every 'Pn maps M ><1 a JR. into M and any cluster point of
the sequence { CfJn} is a conditional expectation onto M. D

The dual action & of IR on M ><1 a JR. is implemented by 10 μ on 1t 0 L^2 (JR.),
where μ(s) is the unitary operator on L^2 (JR.) given by multiplication by e-ist:
(μ(s)e)(t) = e-iste(t) for every s E JR. and e E L^2 (IR). This indeed gives rise
to an JR.-action as we have

(10 μ)(s)(v 0 .A)(t)(l 0 μ)(s)* = e-ist(v 0 .A)(t)

for every s, t E JR. and

(10 μ)(s)(a 01)(10 μ)(s)* =(a 01)

for every s E JR. and a E M. It follows that M ><10: IR is the von Neumann
subalgebra of JBl(H 0 L^2 (JR.) 0 L^2 (JR.)) generated by

Takesaki's duality theorem states

Theorem 9.3.7. M ><la JR. ><10: JR.~ M ® JBl(L^2 (JR.)).