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].
3
http://dx.doi.org/10.1090/cbms/076/02
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