Chapter 15
Classification of Group
von Neumann Algebras
In recent years, C* -tensor product theory and amenable actions of groups
have joined forces to provide some deep results related to the classification
of group von Neumann algebras. For example, it is now possible to recover
Ge's celebrated result that free group factors are prime ([68]) from tensor
product/amenable action techniques; in fact, this approach yields a vast
generalization of Ge's result with no additional effort. Another surprising
application deals with the classification of tensor products (or free products)
of certain von Neumann algebras. Our goal is to explain these applications
in a unified and succinct way.
Throughout this chapter, we make the blanket assumption that all groups
are countable and von Neumann algebras have separable predual.15.1. Subalgebras with noninjective relative cornrnutants
Definition 15.1.1. Let r be a group and g be a family of subgroups of r.
We say a subset n of r is small relative to g if it is contained in a finite
union of sAt's, wheres, t Er and A E Q. (Here sAt ={sat: a EA} c r.)
Let c 0 (r; Q) c £^00 (r) be the C*-subalgebra generated by functions whose
supports are small relative to Q. More intuitively, for a net (si) in r, we
write si ~ oo/Q if si tf:. sAt eventually^1 for every s, t Er and A E Q. Hence,
for f E .e^00 (r) we have that f E co(r; Q) 9 {x E r : lf(x)I > c} is small
relative tog for every c > 0 9 lim 8 _, 00 ;g f(s) = 0.
(^1) Vs, t, VA, :lio such that Vi we have the implication i:;::: io =*'Bi tf_ sAt.
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