2019
DOI: 10.1007/s00208-019-01939-9
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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 automorphi… Show more

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Cited by 10 publications
(23 citation statements)
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“…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.…”
Section: Resultsmentioning
confidence: 82%
See 2 more Smart Citations
“…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.…”
Section: Resultsmentioning
confidence: 82%
“…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…”
Section: Resultsmentioning
confidence: 99%
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“…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).…”
Section: Almost Periodic Statesmentioning
confidence: 99%
“…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]. ??…”
Section: Class S and Unique Prime Factorizationmentioning
confidence: 99%