VON NEUMANN ALGEBRA S 5 conjugacy classe s are infinite. Thu s the suppor t o f / i s the identit y s o that M i s a factor. Cal l it vN(T) . One may see quickly that this factor i s not as in example b) of § 1.4 by ob- serving that th e linear function tr(£/(y) w ) = /(identity) ha s the propert y Xv(ab) = tr(ba) an d i s not identically zero. I t is simple to show that n o such function exist s on 3B{2tf) unles s d i m ^ oo. b) Th e previou s exampl e wa s a n exampl e o f a very genera l constructio n called th e crossed product, wher e on e begin s wit h a vo n Neuman n algebr a N o n %? and a group Y actin g b y automorphism s o n T V (in exampl e a) , N = C) an d one forms a von Neumann algebra M = N x T (o n %?'®f 2 (T)) generated by u y = id 8uy an d an action of N o n ^S/ 2 (T). Al l elements of N x i T ca n be represented a s sums J2 ?er xyuy, x y e N , an d u y xu~l = y(x) (the action of y o n x) fovxeN.ll i s then trivial to show that the following conditions together suffice t o imply that N xT i s a factor . (i) Th e actio n o f T i s "free", i.e. , xy - yy(x) fo r al l x e N implie s y = 0 o r 7 = 1 . (ii) Th e algebra of fixed points for T i s a factor . Crossed products may also be formed by continuous (locally compact) groups, but they are algebraically less transparent. c) Le t u s giv e a n importan t exampl e o f th e previou s construction . Th e group will be Z an d N wil l be L°°(S l ). Th e generator of Z wil l act by an irrational rotation. A s in example a) there is a trace functional o n L°°{S l ) x i Z given on ^2 nez fnun b y f s i f Q (8) dd . Thi s example ca n obviously be varied by replacin g Z b y an y discret e grou p an d S b y an y finite measure space , provided th e group action preserve s the measure an d is free an d ergodic . I t was recognized very early on that in this situation the crossed product algebra depends only on the equivalence relation defined o n the measure space by the orbits o f th e grou p action , indee d tha t i t i s possibl e t o defin e th e crosse d product algebr a give n onl y th e measur e spac e an d th e equivalenc e relatio n (with countable equivalence classes) . Fo r the definitive treatmen t se e [FM]. d) The G.N.S. construction provides a n elementary bu t usefu l wa y to pass from a *-algebra which is not necessarily complete to a von Neumann algebra. The necessary data are a *-algebra A an d (p : A - C wit h (p{a*a) 0. On e then forms a Hilbert space by defining a not necessarily definite inner product on A b y (a , b) = p(b*a). Th e Hilbert spac e 3? i s then the completion o f the quotient of A b y the kernel of this form. Unde r favorable circumstance s (such as if A i s a C*-algebra) , A wil l act on £? b y left multiplication . Thi s representation o f A i s called the G.N.S. representation. Th e von Neuman n algebra generated by the image of A i n this representation should be thought of as a completion of A wit h respect to cp . In general, it is difficult t o say if the G.N.S completion i s a factor o r not. On e often meet s surprises where A has trivial center but its completion doe s not.
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