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 x£

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 y£ 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 .