1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

410 15. Group von Neumann Algebras

We primarily consider p in the case where P = B' n pMp for a pro-
jection p E JJ1 and a diffuse abelian von Neumann subalgebra B C pMp
(meaning B has no nonzero minimal projections). Every diffuse abelian von
Neumann algebra B with separable predual is *-isomorphic to L^00 [0, 1] and
hence is generated by a single unitary element u 0 EB (e.g., u 0 (t) = e^2 7rit).
Fixing such a generator, we define a c.p. map WB from JB(7i) into JB(p7i) by
1 n
WB(x) = ultraweak-lim-L u~xu 0 k,
n n k=l

where the limit is taken along some fixed ultrafilter. It is not hard to see
that WB is a (nonunital) conditional expectation onto B' n JB(p7i) and that
WBlpMp is a trace-preserving conditional expectation from pMp onto B' n
pMp. By uniqueness of the trace-preserving conditional expectation, one
has WB(a) = Ep(pap) for every a EM. It follows that

'11 B (L akbk) = L Ep(pakp )bkp = <I> P(L ak Q9 bk)

k k k

for ak E M and bk E M'.

Proof of Theorem 15.1.5. By contradiction, suppose that the conclusion
of the theorem is not true. Then, by Corollary F.14, there is a diffuse abelian
von Neumann subalgebra BC N such that B does not embed in L(A) inside
M = L(r) for any A. We will use Theorem F.12 with A= L(A). Forthis, ob-
serve that XA E £^00 (r) c JB(L^2 (M)) is nothing but the orthogonal projection
eA onto L^2 (A) and hence XsA = >..(s)eA>..(s)* E (M, A)+ with Tr(XsA) = 1.
It follows that WB(XsA) is a positive element in p(M, A)p n B' such that
Tr(wB(XsA)) ::::; 1. By assumption and Theorem F.12, WB(XsA) = 0. Since
p(I') is in the multiplicative domain of WB, this implies that WB(XsAt) = 0
for every s, t Er and A E g, or equivalently, IK(r; g) c ker WB· Hence, for
the u.c.p. map 0 given in Lemma 15.1.4, one has p = WB o e and p is
min-continuous on CHr) 0 c; (r). Injectivity of p = B' n pM p now follows
from Proposition 15.1.6. D

Exercise

Exercise 15.1.1. Prove the claim made in Remark 15.1.3. Here is a hint:
Let { e} = Eo C E1 c E2 · · · be an increasing sequence of finite symmetric
subsets of r with LJEn = r. Find μn for En and c = l/n. Define relatively
small sets Dn inductively by

Dn = LJ {x: llμn(sxt) - s.μn(x)ll 2: 1/n} U EnDn-1En.

s,tEEn

Set lxl = min{n: x E Dn} and μ(x) = lxl-^1 2:~~ 1 μn(x).