Classification of actions of discrete Kac algebras on injective factors

Abstract

We will study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. We will construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, we will show that the Connes-Takesaki module is a complete invariant.2000 Mathematics Subject Classification. Primary 46L65; Secondary 46L55. 1 4 a ∈ M, the functionals ϕa and aϕ in M * are defined by ϕa(x) := ϕ(ax) and aϕ(x) := ϕ(xa) for all x ∈ M. We denote by U(M), Z(M) and W (M) the set of all unitary elements in M, the center of M and the set of faithful normal semifinite weights on M, respectively. For a positive φ ∈ M * , we set |x| φ := φ(|x|),Denote by Aut(M) the set of automorphisms on M. For α ∈ Aut(M) and ϕ ∈ M * , we let α(ϕ) := ϕ • α −1 . Note that Ad u(ϕ) = uϕu * for a unitary u, where Ad u(x) := uxu * for x ∈ M.Let H ⊂ M be a subspace. We say that H is a Hilbert space in M if H ⊂ M is σ-weakly closed and η * ξ ∈ C for all ξ, η ∈ H [52]. Then the coupling ξ, η := η * ξ gives a complete inner product on H . The smallest projection e ∈ M such that eH = H is called the support of H . If the support of H equals 1, then we have the endomorphism ρ H defined by ρ H (x) = i v i xv * i , where {v i } i is an orthonormal basis. We say that an endomorphism σ is inner when σ = ρ H for some Hilbert space H with support 1.Next we recall theory of endomorphisms on a factor. Our standard references are [22,23,40,41]. We denote by End(M) and Sect(M) the set of normal endomorphisms and sectors on M, that is, Sect(M) is the set of unitary equivalence classes of endomorphisms on M. For two endomorphisms ρ, σ ∈ End(M), we letFor a factor M, we denote by End(M) 0 the set of endomorphisms with finite Jones-Kosaki index. For ρ ∈ End(M) 0 , E ρ denotes the minimal expectation from M onto ρ(M) in the sense of [20]. We define the standard left inverse of ρ by1.2. Canonical extensions. In [23], Izumi introduced the canonical extension of an endomorphism, which plays an important role in this paper.Let M be a factor and M the core of M as defined in [16, Definition 2.5]. The core M is the von Neumann algebra generated by M and a one-parameter unitary group {λ ϕ (t)} t∈R , ϕ ∈ W (M), satisfying the following relations:The action θ is called the dual action, and the flow of weights of M means the restriction of θ on the center Z( M).Definition 1.1 (Izumi). Let ρ be an endomorphism on a factor M with finite index. Then the canonical extension ρ of ρ is the endomorphism on M defined byfor all x ∈ M, t ∈ R and ϕ ∈ W (M).Note that ρ commutes with the dual action θ. Conversely, if β ∈ End( M ) is commuting with θ, then the restriction β| M gives an endomorphism on M because of M = M θ , the fixed point algebra by θ. In the following lemma, we study a relationship between β and β| M . On a probability index for an inclusion of von Neumann algebras, readers ar...

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