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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