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 ÷ ^)(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  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).
Previous Page Next Page