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~ X v) (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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