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