Lecture 1 . von Neumann Algebra s 1.1. Three topologies on 3\%f). I f %* is a complex Hilbert space with in- ner product ( , ) , the norm topology on the *-algebra S§{^) o f all bounded linear operators on %? is defined b y the norm: ||*| | = su p ^% ||x^||. In finite dimension s x i s a matri x an d \\x\\ ma y b e calculate d a s th e largest eigenvalu e o f x*x . Th e sam e i s tru e i n infinit e dimension s i f w e replace "larges t eigenvalue " b y "spectra l radius" , wher e th e spectra l radiu s of a n operato r a i s sup{|/, | : X - a i s not invertible} . I f w e consider %? L2([0, 1] , dx), L°°([0 9 1] , dx) act s on JT b y pointwise multiplicatio n an d the nor m of f e L°° i s th e essentia l su p o f \f\ . Thu s th e continuou s functions C([0 , 1] ) for m a nor m close d subalgebr a o f L°°([0, 1] , dx) o n %f. (Not e that the choice of ([0 , 1] , dx) i s inessential. Th e same things are true for any compact space and measure you are likely to think of in the next ten minutes. ) The strong topology on 38{2l?) is that defined by the seminorms x *- \\x£\\ as £ run s through %? . Thu s a sequence (o r net if you must) x n converge s to x if f x n £ converge s t o xl i n %? for al l £ e %? . Th e stron g topolog y is much weake r tha n th e nor m topology . I n fac t w e will soo n se e that , fo r the exampl e o f L°°([0 , 1] ) an d C([0 , 1] ) actin g o n L 2 ([0, 1]) , C([0 , 1] ) is actually strongl y dens e i n L°°([0 , 1]) . T o se e a sequenc e tha t converge s strongly without converging in norm, let x n b e the characteristic function o f [0, l/n] viewe d as an element of L°° . Obviously x n tend s strongly to zero, but ||jc j = 1 for al l n. The weak topolog y o n &&{%?) i s tha t define d b y th e seminorm s x i- |(*£,*/)| a s £ an d r\ ru n throug h %f. Th e Cauchy-Schwar z inequalit y shows that the strong topology is stronger than the weak topology. I n fact th e weak topology i s so weak that th e unit ball of 3§{%*) is weakly compact which i s often ver y useful. Probabl y th e simples t exampl e o f a sequence of operators tendin g weakl y bu t no t strongl y t o zer o i s th e sequenc e e ind i n L°°(Sl) (o n L (5 1 )), whic h b y Fourie r serie s i s th e sam e a s th e obviou s shift operato r on / (Z) . I http://dx.doi.org/10.1090/cbms/080/01
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