OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS 11
/ · V = f ì-a.e . for all V e [R]} . In particular, sf is a factor if and only
if R is ergodic, and sf - B(H) in this case.
In order to construct a more interesting von Neumann algebra from a non-
singular equivalence relation R on (X, S?, ì) we follow [FelMl] (see also
[Verl]) and set Ç =
L2(R,
ì£
])
(cf. (1.1)). For every h e Ó°°(×,ì) we
obtain multiplication Operators Mh, M'he B(H) by setting (Mhf)(x, x) =
h(x)f(x, x), (M'hf)(x, x) = / ( * , x)h(x), f e Ç, and we put ^#(R) =
{ ^ : Ë Å Ã ( É , / / )}
a n d ^ ' ( R ) = {M^AeL°°(*,//)} . For V e [R] (Ex-
ample 1.2(1)) we define unitary Operators LV,L'V e B(/f) by
(Lvf)(x, x) =
f(V~lx,
÷) and (À//)(÷,÷')
ê
= / ( x , F~V)-/
R / |
(F~ V ,
x') ' , f e Ç. Then F Lv and F L'v are homomorphisms from [R]
into B{H), and
L"1
· ËÔ(R) · L
F
= L'v-i · ËÔ(R) · L'F = Jt{R) for every
Ê Å [R]. The algebra j/(R) = (^f(R) U {LK: F e
[R]}),/
is called the von
Neumann algebra ofthe equivalence relation R. If R is ergodic, then j/(R)
is a factor, ^f (R) c J / ( R ) is maximal Abelian, and sf(R) = Jf(JP(R))" ,
where Jf(Jf(R)) is the normalizer of Jf(R)
,14
Since the (isomoöhism dass of the) von Neumann algebra sf(R) is
unaffected if we replace ì by an equivalence measure í ~ ì we can assume
without loss in generality that ì(×) = 1 . The unit vector ù e Ç given by
ù(÷, ÷') = ä÷
÷
16
is cyclic under s/(R) (i.e. j/(R)a is dense in H), and
we set, for every Á €J/(R) ,
(1.10) çì (Á) = (Áù,ù).
Then ç : J / ( R ) C is a (faithful) State, i.e. a bounded linear functional
such that ç (l) = 1 and çì (Á*Á) 0 whenever 0 Ö Á e J/(R), and
çì Ç) = f ÇÜì for all Á e L°°(X, ì). If R preserves ì , then ç is a
(normalized) irace on s/(R), i.e. ^(Aß) = ç (ÂÁ) for all ^ , i G ^ ( R ) .
If ì is an infinite R-invariant measure on X we can formally define ù and
ç as above, but ù £ Ç, and f / (Ë) cannot be defined for all elements of
j / ( R ) . The domain ¼(ç ) = e jif(R): ç (Á*Á) oo} of ç is a dense
subalgebra of s/(R), çì (ÁÂ) = çì {ÂÁ) for all Á, Â e ¿(ç
ì
), and çì is a
semifinite trace on J / ( R ) .
1.3.
EXAMPLES.
(1) The von Neumann algebra of á free group action. Let
The normalizer of ^#(R) is the set of all unitary Operators t/GJ/(R) with C/~ -^#(R) ·
(7 = ^f(R). One checks easily that {MA , M^:/? L°°(*, ì)} " = L°°(R, ì
ê
), j/(R) =
(ËÔ'(Ê ) U {L'K: Ê e [R]})" C J^(R)' and j/(R) C (^'(R))' . From this it is clear that s/(R)
is a factor with nontrivial commutant, .#(R) c J / ( R ) is maximal Abelian, and J / ( R ) =
Jf(J?{R))" .
Á homomorphism Ö:^/ & of C*-algebras is a linear map satisfying Ö(1) = 1 ,
Ö(Á*) = Ö{Á)* , and Ö(ÁÂ) = Ö(Ë)Ö(AE?) for all /1, 5 G i , and an isomorphism is a
bijective homomorphism.
Sa n is the Kronecker delta: SQ * = 1 if á = /?, and 5Q « = 0 otherwise.
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