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 %?
1], dx), L°°([0
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
converge s
to x if f x 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
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
b e the characteristic function o f
[0, l/n] viewe d as an element of L°° . Obviously x
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
i n
(o n L (5
)), 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) .
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