4

OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS

came the investigation of those properties of a nonsingular action g — • Ô

of a

countable3

group G ona measure space (X, 5?, ì) which are carried

by the orbit structure, independently of the way in which these orbits are

generated.

The theory of Operator algebras not only led to the study and partial Classi-

fication of orbit Spaces (or countable equivalence relations), but also provided

many of the ideas central to the investigation of the properties carried by the

orbit structure. These ideas have since been stripped of their origins and can

be discussed and used without any reference to (or knowledge of) Operator

algebras, but I shall try to offer a glimpse of the context they originated from

in order to provide motivation, and to point the way to an area where there

are likely to be many more hidden treasures.

1.1.

DEFINITION

[FelMl]. Let (X, S*) be a Standard Borel space. ABorel

set R c l x l is a (countable or discrete) Borel equivalence relation on X

if R is an equivalence relation, and if, for every ÷ e X, the equivalence

dass R(jt) = {x e X: (x, x) G R } of × is countable. Á Borel equivalence

relation R on X isfinite if R(x) is finite for every ÷ e X. If S is a second

Borel equivalence relation on X then S is a subrelation of R if S(JC) C R(JC)

for every ÷ e X.

Let R be a Borel equivalence relation on X and let ì be a measure

on 5?. Then R is (ì-)nonsingular if ì(¸ß(Á)) = 0 for every Á e ¥

with ì(Á) = 0, where R{A) = \JxeAR(x) denotes the Saturation of a set

Á c X. Á nonsingular equivalence relation R is transitive if ì(×\Ê(÷)) = 0

for some ÷ e X and intransitive otherwise. An intransitive equivalence

relation R is (properly) ergodic if ì(Ê(Á)

€)

= 0 whenever Á e S? and

ì(Á) 0. Two nonsingular equivalence relations R, S on (X, ¥*, ì) are

equal (mod ì) if there exists a //-null set N G ^ such that R\(N ÷ Í) =

S\(N ÷ Í) . Two nonsingular equivalence relations R and S on measure

Spaces (X, 5?, ì) and (Y, &, v) are isomorphic if there exists a measure

space isomorphism ø:(×, S*, ì) — • (Õ, T, í) such that (ø ÷ ^)(R) =

S ( mod í) , and the map ø is an

isomorphism5

of R and S.

1.2.

EXAMPLES.

(1) Equivalence relations and group actions. Let T:g —

Ô be a nonsingular action of a countable group G on (X, S?, ì) . Then

The restriction to actions of countable groups and to countable equivalence relations is

inessential, but has the advantage of technical simplicity. Ingredients for extending the theory

to actions of locally compact, second countable groups and the associated equivalence relations

can be found in [FelHM] and [FelR].

4Corollary

2 in [Kur, §39, VII] implies that R(^) e 5? for every AeS*.

If R = S then ø is an automorphism of R, and ø is an inner automorphism of R if

wem.

If (X

y

y, ì) is a measure Space, a surjective Borel map V: X —• X is a nonsingular

endomorphism of X if, for every Â e S^, ì(Â) = 0 if and only if ì(Õ~ Â) = 0 (in order for

ì¥~ to be ó-finite it may be necessary to replace ì by an equivalent probability measure).