14.1. Defi.nition and examples 395converges to if in the point-ultraweak topology. This implies the continuity
of if. D
Corollary 14.1.5. All of the following statements are true.
(1) Every injective (semidiscrete) von Neumann algebra is weakly ex-
act.
(2) If M is a weakly exact von Neumann algebra, then M ® B(H) is
weakly exact.
(3) A von Neumann algebra M is weakly exact if and only if its com-
mutant M' is.
(4) If Mis a weakly exact von Neumann algebra and G is an amenable
{locally compact) group which acts on M, then M ><I G is weakly
exact.
(5) A von Neumann subalgebra N C M is weakly exact provided that
M is weakly exact and there exists a normal conditional expectation
from M onto N. In particular, every von Neumann subalgebra of
a finite weakly exact von Neumann algebra is weakly exact.The proofs of these facts make nice exercises, so we leave them to you.
It is unknown whether or not every conditional expectation from a von Neu-
mann algebra M onto a von Neumann subalgebra N can be approximated,
in the point-ultraweak topology, by normal c.p. contractions from M into
N. However, it is known to be true "up to Morita equivalence" [4]. Hence
the last fact of the above corollary holds without the normality assumption
on the conditional expectation.
Ifr is an exact discrete group, then the group von Neumann algebra L(r)
is weakly exact since it contains the weakly dense exact C*-algebra C~(r).
The converse is also true, though we postpone the proof until Section 14.2.
Theorem 14.1.6. For a discrete group r, the group von Neumann algebra
L(I') is weakly exact if and only if r is exact.
More generally, if a is a measure-preserving action of r on a probability
space (X, μ),then the crossed product von Neumann algebra L^00 (X, μ) ><1 r
is weakly exact if and only if r is exact. Indeed, if r is exact, then the C -
crossed product L^00 (X, μ) ><lrI' is weakly dense in L^00 (X, μ) ><1 rand is exact
since L^00 (X, μ) is an exact (nuclear) C-algebra; the converse follows from
the previous theorem, together with the fact that weak exactness passes to
subalgebras of finite von Neumann algebras. In particular, exactness is pre-
served under so-called measure equivalence (cf. [185, Proposition XIII.2.16]
and [67]), as this implies stable isomorphism of the corresponding crossed
products.