4
VON NEUMANN ALGEBRAS
L°°([0, 1]) o n L
2([0,
1]) th e correspondin g decompositio n o f L
2([0,
1])
would b e /
x
^{X)dx(X) wher e W{k) = C . Th e whol e algebr a M re -
spects thi s decompositio n an d w e en d u p wit h a notio n o f direc t integra l
of von Neumann algebras : M = M{/J) dk o n ^(k) dk , th e whole de-
composition being essentially unique. Th e individual M(A)' s will have trivial
center (t o get a feel fo r this , work i t out in finite dimensions). Thu s any von
Neumann algebr a is the direct integra l o f ones with trivia l center .
Although th e technica l detail s o f thi s theory ar e rathe r messy , an d i t ca n
usually be avoided by "global" methods, the direct integra l decomposition i s
tremendously helpful i n trying to visualize a von Neumann algebra on a basic
level. O f cours e on e doe s no t ge t an y furthe r tha n th e A f (A)'s with trivia l
center.
DEFINITION.
A von Neuman n algebr a M whos e cente r i s just th e scala r
multiples of the identity i s called a factor.
EXAMPLES.
a) 3B{%T) i s a factor .
b) I n finite dimensions a factor wil l always be of the form 3&ffl) ® C id
on ^ g J?. Thi s i s als o tru e i n infinit e dimension s provide d th e
factor i s isomorphic, a s an abstract algebra , to some 3B{S^).
Example b) explains the name "factor"—such factor s correspond to tensor
product factorization s o f the Hilbert space . Th e remarkable fact , discovere d
by Murra y an d vo n Neuman n i n thei r work s [MvNl , 2 , 3 ] i s tha t no t al l
factors ar e lik e this , an d indeed , a s w e shal l see , i t i s no t ver y difficul t t o
construct examples .
1.5. Example s of factors, a ) Let T b e a discrete group (e.g., the free group
on two generators) al l of whose conjugacy classe s are infinite, excep t tha t o f
the identit y (w e wil l cal l suc h group s i.c.c) . Le t y u denot e th e left -
regular representatio n o f T o n / (r) . A s matrice s o n / (T) wit h respec t
to th e obviou s basis indexe d b y y e T , th e u , an d henc e al l element s o f
alg({w^}) ar e of the form x
v
= f{y~
Xv)
(forgiv e m e if the inverse is in the
wrong place) fo r som e function / o f finite support o n T . Th e same is true
for wea k limit s o f suc h operator s excep t tha t / wil l n o longe r b e o f finite
support. However , applying the operator to the basis element for the identity
we see that / i s in f
2
. It is thus convenient and accurate to write element s
of M = {u y}" a s sum s S
y r
/(7)w
y
wher e / e f
2
(althoug h no t al l I
2
functions defin e element s of M). Th e sense of convergence o f the sum will
be clear later on. I n any case, in order that 5Z„
€r
f{y)u belon g to the center
of M , i t must commute with u
u
fo r all v , which implies f{vyv~ ) = f(y) ,
i.e., / i s constant o n conjugac y classes . Bu t / i s in / an d al l nontrivia l
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