Ergodic Transformation Groups and the Associated von Neumann Algebras

Abstract

We now begin our systematic study of the von Neumann algebra associated with an ergodic transformation group of a standard measure space. We touched the subject lightly in Chapter V when we constructed a factor of type III. As ergodic transformation groups of a standard measure space appear almost everywhere in mathematics, the construction provides a rich family of examples as well as the opportunity of applications of the theory of von Neumann algebras. Thus this is an important area of operator algebras. In §1, we will define the von Neumann algebra R(G, Ω, µ) associated with an ergodic transformation group G of a standard measure space {Ω, µ} as the crossed product L∞(Ω,µ) x G. We then relate the maximal commutativity of L∞ (Ω, µ) in R(G,Ω, µ) to the freeness of the action of G. The ergodicity of G is then shown to be equivalent to the factoriality of R(G,Ω, µ). The type question of the factor R(G, Ω, µ) is shown to be directly linked to the existence of invariant measures. Section 2 is then devoted to the technique that allows us to construct a factor from a non-free but ergodic transformation group. It will be shown that the main issue in constructing a factor out of an ergodic transformation group is not the group itself nor the action but rather its orbit structure. This motivates us to look at the equivalence relation associated with an ergodic transformation group; thus the theory of measured groupoids in §3. This shift of our focus from groups to groupoids gives us a great deal of flexibility in the structure analysis of the associated factor in later chapters. The merit of this flexibility can be seen already in the study of amenable groupoids in §4. Whilst we cannot approximate an amenable group by a sequence of finite groups, an amenable principal groupoid is indeed approximable by a sequence of “finite” groupoids as seen in Theorem 4.10. Later, in Chapter XVI, we will see that the amenability of a factor, which will be defined following the analogy with the group case, will imply the possibility of approximation by finite dimensional subalgebras.