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 .

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