Structure of bicentralizer algebras and inclusions of type III factors

2018/04/16 by Hiroshi Ando, Uffe Haagerup, Ando, Hiroshi +5
Mathematics · #46L10 #46L30 #46L36 #46L37 #46L55 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1804.05706

openalex publication_date 2018/04/16 · openalex created_date 2019/12/05 · openalex updated_date 2026/07/28

Abstract

We investigate the structure of the relative bicentralizer algebra \rm B(N ⊂ M, φ) for inclusions of von Neumann algebras with normal expectation where N is a type \rm III₁ subfactor and φ∈ N_* is a faithful state. We first construct a canonical flow β^φ: \mathbf R^*₊ \curvearrowright \rm B(N ⊂ M, φ) on the relative bicentralizer algebra and we show that the W^*-dynamical system (\rm 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 \rm 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.

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