Locally Compact Separable Abelian Group Actions on Factors with the Rokhlin Property

Shimada, Koichi

doi:10.4171/prims/137

Cited by 1 publication

(1 citation statement)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[81]. On the basis of this work, Shimada also studied actions of R-actions and locally compact abelian group actions on von Neumann algebras in [82,83,84].…”

Section: Acknowledementmentioning

confidence: 99%

Actions of groups and quantum groups on amenable factors

Masuda

Tomatsu

2017

Sugaku Expositions

View full text Add to dashboard Cite

is C, then M is called a factor. Since every separable von Neumann algebra is a direct integral of factors, factors are the most important objects in the study of von Neumann algebra.We next explain group actions. We denote the automorphism group of M by Aut(M ). Namely, it is the set of all bijections which preserves the * -algebra structure of M . 1 Let G be a locally compact group (in this exposition, we only treat second countable topological groups). Let α be a group homomorphism α : G → Aut(M ). If α satisfies the following continuity condition, then we will say α is an action, and we often write α : G M :If G is discrete, then this continuity condition holds automatically. Denote U (M ) := {u ∈ M | u * u = 1 = uu * }, and we say that an element in U (M ) is a unitary. Each u ∈ U (M ) defines the automorphism Ad u(x) := uxu * , which is called an inner automorphism, and we denote the set of all inner automorphisms by Int(M ). 2 From the equality αDefinition 1.1. Let M be a von Neumann algebra, G be a locally compact group, and α be an action of G on M .(1) A continuous map v :(2) An α-cocycle v is said to be a coboundary if there exists w ∈ U (M ) with v g = wα g (w * ) for all g ∈ G.(3) Let v be an α-cocycle. Then Ad v g • α g is also an action. We will call this action the cocycle perturbation of α by v.Definition 1.2. Let M be a von Neumann algebra, G be a locally compact group, and α, β be actions of G on M .(1) We will say α and β are conjugate if there exists θ ∈ Aut(M ) such that(2) We will say α and β are cocycle conjugate if there exists an α-cocycle v such that Ad v g • α g and β g are conjugate. When this is the case, we denote by α ∼ β.Usually, the cocycle conjugacy is too strong for us to classify actions, and it is natural to classify by cocycle conjugacy in many situations. Thus our purpose is to solve the following problem. Problem 1.3. For a given von Neumann algebra M and G, classify all G-actions by cocycle conjugacy equivalence.In fact, such classification should be possible when M and G are "amenable". In the non-amenable case, it is known that we cannot obtain classification. (See [12] for example. We refer to [56] for the study of non-amenable factors and non-amenable groups.)At the end of this subsection, we explain how we can construct a new von Neumann algebra, called a crossed product von Neumann algebra, from a group action α : G M . This construction plays an important role throughout this

show abstract

“…[81]. On the basis of this work, Shimada also studied actions of R-actions and locally compact abelian group actions on von Neumann algebras in [82,83,84].…”

Section: Acknowledementmentioning

confidence: 99%