VON NEUMANN ALGEBRAS 3 EXAMPLES. i) Th e algebr a SB (%?) itsel f i s certainl y closed , thu s a von Neuman n algebra. ii) Th e algebra L 00 ([0, 1 ], dx) i s easily shown to be its own commutant, thus a von Neuman n algebra . iii) I f G i s a group an d g - u i s a unitary representatio n o f G , the n the commutant {u } ' i s a von Neumann algebra . iv) I f ^ i s finite-dimensional, i t is not too hard t o see that a von Neu - mann algebr a M i s just a direct su m of matrix algebras correspond - ing to som e orthogona l decompositio n 2f? = 2f? x the matrices in M wil l look like »%l s o tha t dim^f dim^ x I J e V ^ J d i m ^ where the x t 's ar e matrices. v) I f M o n %f an d N o n ^ ar e vo n Neuman n algebra s ther e ar e obvious notions of direct sum M®N o n %* (§3? an d tensor product A/®AT on ^ ® J f . We list some important fact s abou t von Neumann algebras . 1 Th e se t o f al l projection s o f a vo n Neuman n algebr a M form s a complete (orthomodular ) lattice . M i s generated b y it s projection s since it contains th e spectral projections o f any selfadjoint element . 2) Abelia n von Neumann algebras.are completely understood. A s well as example ii) above there is /°°(N ) o n / (N ) an d obvious reductions and combination s wit h exampl e ii) . On e mus t allo w som e kin d o f "multiplicity" as can be seen in finite dimensions. Bu t on a separable Hilbert spac e that i s the whole story. Probabl y th e best wa y to dea l with the multiplicity question i s to relegate it to the spectral theore m and state , a s vo n Neuman n did , th e structur e theore m fo r abelia n von Neumann algebra s as the fact tha t they are generated by a single selfadjoint operator . 3) vo n Neuman n algebra s ca n b e abstractl y characterize d a s C* - alge- bras which ar e duals as Banach spaces . Se e [Sa]. 1.4. Factors . Th e cente r Z(M) o f a vo n Neuman n algebr a i s abelian . So b y fac t 2 o f §1. 3 w e kno w everythin g abou t it . I n finite dimensions it woul d b e a direc t su m o f copie s o f C , on e fo r eac h summan d i n th e decomposition o f exampl e 4 o f §1.3 . I n general , usin g th e spectra l the - ory, vo n Neuman n define d ([vN2] ) (i n th e separabl e situation ) a notio n o f "direct integral " o f Hilber t space s J ® ^(k)d\i{k) s o that , fo r instance , fo r
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