1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

296 9. Local Reflexivity

Proof. For simplicity, let M = M ><1 °' JR. ><1 & R Since ,\ and μ are unitarily

equivalent via the Fourier transform, M is spatially isomorphic to the von

Neumann algebra generated by

(1@ μ@μ)(JR.), (v@ ,\@ l)(JR.) and M@ Cl @Cl.

If U is the unitary operator on L^2 (JR.) @ L^2 (JR.) ~ L^2 (JR. x JR.) given by

(Ue)(s, t) = e(s - t, t), then one has U(μ@ μ)(r)U* = (μ@ l)(r) and

U(,\@ l)(r)U* = (,\@ l)(r) for every r E JR.. Indeed,

and

(U(μ 0 μ)(r)u*e)(s, t) = ((μ 0 μ)(r)u*e)(s - t, t)

= e-ir(s-t)-irt(u*e)(s _ t, t)

= e-irse(s, t)

= ((μ 0 l)(r)e)(s, t)

(U(.\ 0 l)(r)ue)(s, t) = ((.\ 0 l)(r)ue)(s - t, t)
= (ue)(s - t - r, t)
=e(s-r,t)
= ((.\ 0 l)(r)e)(s, t).
It follows that Mis spatially isomorphic to the von Neumann algebra gen-
erated by
(1@ μ@ l)(JR.), (v@ ,\@ l)(JR.) and M@ Cl@ Cl.
We note that the von Neumann algebra generated by the first and third
items is M ® £CXl(JR.) @Cl. Let V be the unitary operator on 1i@ L^2 (JR.)
associated with v - i.e., (Ve)(t) = v(t)e(t) fore E L^2 (JR., 1i) ~ 1i@ L^2 (JR.).
It follows that V(v@ ,)(r)V = (1 @,)(r) for r ER
We claim that
V(M ® L^00 (JR.))V = M ® L^00 (JR.).
Since V acts "diagonally,'' V commutes with Cl@ L^00 (JR.). On the other
hand, V(a@ l)V = 1ra(a) for every a E M, where 1ra(a) is given by
(7ra(a)e)(t) = ct-t(a)e(t). It follows that 7ra(M) c L^00 (JR., M) ~ M®L^00 (JR.).
Hence, V(M ® L^00 (JR.))V c M ® L^00 (JR.). The converse inclusion follows
from V(M@ Cl)V c M ® L^00 (JR.), which can be shown by a similar argu-
ment. This proves that V*(M ® L^00 (JR.))V = M ® L^00 (JR.).

Consequently, M is spatially isomorphic to the von Neumann algebra
generated by
M ® L^00 (JR.)@ Cl and (1@ ,\@ l)(JR.).

Finally, we observe that the von Neumann algebra generated by L^00 (JR.) and
,(JR.) is llll(L^2 (JR.)) since L^00 (JR.)' n ,(JR.)'= Cl. D