294 9. Local Reflexivityif its comnmtant is semidiscrete (Corollary 3.8.6) and we already know that
if A is nuclear and 7r: A --+ JIB(H) is any representation (e.g., the univer-
sal representation), then the commutant 7r(A)' is an injective von Neumann
algebra (Exercise 3.6.4). Hence we'll be finished once we establish the fol-
lowing remarkable theorem.
Theorem 9.3.3. A von Neumann algebra is semidiscrete if it is injective.^3
We first prove this when M is semifinite - i.e., when it has no summand
of type III - and then reduce the general case to the semifinite case via
modular theory.
Theorem 9.3.4. Every injective semifinite von Neumann algebra is semidis-
crete.
Proof. It is easily seen that if M = M1 EB M2, then M is injective (resp.
semidiscrete) if and only if both M1 and M2 are injective (resp. semidis-
crete). Since we already proved that type I von Neumann algebras are
semidiscrete (Proposition 2.7.2), we may assume that M = Mu 1 EB Mu=, for
some algebras of type II1 and II 00 , respectively. But Mu= is an increasing
union of type II1 corners (which will be injective whenever Mu= is injective)
and hence Lemma 2.7.1 allows us to assume that Mis of type II1 and has
a normal faithful tracial state r. Since M is injective, the tracial state r is
amenable (Exercise 6.2.1). Thus, by Theorem 6.2.7, the -homomorphism
7r 7 x 7r~P: M@M^0 P--+ JIB(L^2 (M,r)) is continuous. We note that 7r 7 and 7r~P
are normal -monomorphisms and in particular that 7r 7 (M)' = 7r~P(M^0 P)
(Theorem 6.1.4). Thus, by Theorem 3.8.5, the identity map on Mis weakly
nuclear - i.e., Mis semidiscrete. D
Modular theory was one of the highlights of research in the 1970s. We
won't prove the following theorem -it is deep and difficult - so consult [184,
Theorem XII.1.1] if you must.
Theorem 9.3.5. For any von Neumann algebra M there exists an action a
of JR on M (the so-called modular action) such that M ><la JR is a semifinite
von Neumann algebra.
Thus our work will be complete once we show that injectivity is preserved
by JR-crossed products and that semidiscreteness passes from the crossed
product back to M.
An JR-action is a group homomorphism a: JR --+ Aut(M) such that t f-->
at(a) is ultraweakly continuous for every a E M. We may assume that
M C JIB(H) and the action a is implemented by a strongly continuous unitary
(^3) Recall that the converse is easy - see Exercise 2.3.15.