1. Operator Algebras and Dynamical Systems

The connection between Operator algebras and ergodic theory goes back

almost to the origins of the two subjects. In 1936, F. J. Murray and J. von

Neumann [MurNl] introduced the group measure space construction of von

Neumann algebras from group actions on measure Spaces and showed that

these algebras are factors, i.e. have trivial centres, if and only if the group

actions are ergodic. This construction (with later refinements) remains one

of the most important sources of examples of Operator algebras. Á striking

early result in von Neumann algebras was the proof of the uniqueness of

the hyperfinite Ilj-factor in 1943 [MurN2], which implied in particular that

the factors constructed from any two measure-preserving, ergodic automor-

phisms on probability Spaces are isomorphic. Building on earlier work by

E. Hopf [Hop], H. A. Dye investigated the reason for the existence of such

an isomorphism and found that the von Neumann algebra arising from the

group measure space construction only depends on the orbits of the group

(or the Single transformation) acting on the measure space, and that any two

measure-preserving, ergodic automorphisms of probability Spaces have iso-

morphic orbits [Dyel]. In [Dye2] he showed that every measure-preserving

action of a countable Abelian group (cf. [Abel]) on a probability space has

the same orbits as a suitably chosen Single automorphism of the measure

space, thereby providing a proof for an announcement in [MurN2] that every

measure-preserving, ergodic action of an infinite Abelian group on a proba-

bility Space again leads to the hyperfinite Ilj-factor. This emphasis on orbits

rather than group actions was brought to its logical conclusion in [FelMl] and

[FelM2], where the acting group is suppressed completely and the orbit space

is replaced by an equivalence

relation.2

Hand in hand with this development

Á measure Space (X, S?, ì) is Standard if (X, 5?) is a Standard Borel Space (i.e. Borel

isomorphic to a Borel subset of R with the induced Borel structure) and ì is a ó-finite measure

on S? . The elements of S? are the Borel sets. If there is no danger of confusion we sometimes

write {×, ì) instead of {X, S", ì) . All measure Spaces are assumed to be Standard, and all

measures ó-finite.

The study of equivalence relations is a special case of G. W. Mackey's investigation of Virtual

groups [Macl], [Mac2], [Raml]. For a discussion of equivalence relations from a geometric point

of view we refer to [Con5], and topological equivalence relations are discussed in [Ren].

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http://dx.doi.org/10.1090/cbms/076/02