Structure of bicentralizer algebras and inclusions of type $$\mathrm{III}$$ factors

Abstract

We investigate the structure of the relative bicentralizer algebra B(N ⊂ M, ϕ) for inclusions of von Neumann algebras with normal expectation where N is a type III1 subfactor and ϕ ∈ N * is a faithful state. We first construct a canonical flow β ϕ : R * + B(N ⊂ M, ϕ) on the relative bicentralizer algebra and we show that the W * -dynamical system (B(N ⊂ M, ϕ), β ϕ ) is independent of the choice of ϕ up to a canonical isomorphism. In the case when N = M , we deduce new results on the structure of the automorphism group of B(M, ϕ) and we relate the period of the flow β ϕ to the tensorial absorption of Powers factors. For general irreducible inclusions N ⊂ M , we relate the ergodicity of the flow β ϕ to the existence of irreducible hyperfinite subfactors in M that sit with normal expectation in N . When the inclusion N ⊂ M is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when N is amenable.2010 Mathematics Subject Classification. 46L10, 46L30, 46L36, 46L37, 46L55.

On the Relative Bicentralizer Flows and the Relative Flow of Weights of Inclusions of Factors of Type III$_1$

“…If s ∈ Ξ 0 , then we can easily see a k,s ∈ C by Haagerup's theorem B(N, ϕ) = C [5] and the fact lim n→∞ σ ϕ s (x n ) − x n = 0 in the strong topology. (2) The proof is similar to that of [1,Proposition 6.11]. Take (a n ) ∈ ℓ ∞ (N, N) with lim n a n ϕ − λϕa n = 0.…”

“…Moreover we have dim H s = 1, s ∈ Ξ 0 . In this case, N ′ ∩M is described in [1,Theorem 6.9], which can be regard as a generalization of computation in [10], [15]. Namely, fix a unitary V s ∈ H. Then we have…”

“…Since the canonical extensionα g is inner if and only if g ∈ N, we can see that N ′ ∩M = { n∈N a n λ ϕ (ν(n)) * V n | n ∈ N} as in [10], [15], [1,Proposition 6.11].…”

Connes' bicentralizer problem for q‐deformed Araki–Woods algebras

“…Here is another short proof of item (2), by assuming that M is a type III factor with separable predual. By (the proof of) [13,Theorem 3.5], the centralizer of B(M, ϕ) is trivial, while ϕ is almost periodic on B(M, ϕ).…”

Ozawa's class𝒮for locally compact groups and unique prime factorization of group von Neumann algebras

“…In [20, theorems B], Houdayer and Isono provide a unique prime factorization theorem for non-amenable factors satisfying strong condition (AO). A slightly more general version, removing the condition that the unknown tensor product factors N i have a state with large centralizers, was later proved by Ando, Haagerup, Houdayer and Marrakchi in [3,Application 4]. ??…”