2 VON NEUMANN ALGEBRAS
The interplay between the above three topologies is basic to von Neumann
algebras. W e refrain fro m mentionin g the many other topologies around .
1.2. vo n Neumann's bicommutant theorem. Le t us prove the following sim-
plified version of von Neumann's bicommutant theore m (se e [vNl]). W e use
the followin g standar d notation : i f S c 3&{%T) the n S' = {x e &{%?) :
xs = sx fo r al l 5 e S}, an d S" = (S')
f.
THEOREM.
Let S be a subset of 3§\%f) with the following two properties:
a) If xeS then x* e S ;
b) 1 e S ( 1 is the identity operator on 3B[S^)) .
Then alg(S ) is strongly (hence weakly) dense in S" . (alg(S ) is the algebra
generated by S .)
PROOF.
Firs t chec k tha t alg(S ) c S". No w suppos e y e S". Wha t w e
must sho w i s this: fo r an y finite set £ {, ...
n
i n %? , there i s an elemen t
x o f alg(S' ) wit h
t
arbitraril y clos e to y^
i
fo r al l i . Le t u s suppos e a t
first that w e only want t o approximate on e vector y£. Th e trick i s this: le t
V b e the closur e o f the vector subspac e alg(5) ^ an d le t p b e the operato r
that i s orthogonal projection ont o V . Clearl y aV c V fo r al l a e S; s o by
property a) , ap pa. Thu s yp = py sinc e y e S" . S o yV c V . Bu t by
property b), £ e V s o that e alg(5
,)J
, which i s precisely what we wanted
to prove.
The genera l cas e o f €x, ...
n
involve s anothe r tric k whic h i s use d al l
over th e subject : mak e ^ , £2, .. . , £n int o a singl e vecto r o n th e Hilber t
space © "
= 1
^ . The n alg(S 7.) an d y ac t diagonally on ®
n
i=x^ an d we can,
after makin g some matrix calculations to see how commutants behave under
this "amplification" o f %* , repeat the previous argument with f replaced by
©"=i^/ t o conclude the proof. D
NOTE.
I f 1 di d no t belon g t o S th e theore m stil l applie s provide d on e
cuts down to the closed subspace of %? that i s all that S notices .
This beautiful littl e theorem show s that two notions, one analytic (closur e
in the strong topology) and one purely algebraic (being equal to one's bicom-
mutant) are the same for *-subalgebras of g&ffi) containin g 1. It thoroughly
justifies the definition of § 1.3. Not e also that the theorem shows that "strongly
closed" and "weakly closed" are the same thing for a *-subalgebra of 3&{%?).
1.3. (Concrete ) von Neumann algebras.
DEFINITION.
I f %? is a complex Hilber t space , a von Neumann algebra is
a *-subalgebr a M o f SS(^) containin g 1 such tha t eithe r M i s strongl y
(weakly) close d o r M - M" . I f S i s a selfadjoin t subse t o f &(%f) the n
S" i s the von Neuman n algebr a generated b y 5 .
Previous Page Next Page