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 (Ì 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 ) = {Á 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.