# Structure of bicentralizer algebras and inclusions of type III factors

@article{Ando2018StructureOB, title={Structure of bicentralizer algebras and inclusions of type III factors}, author={Hiroshi Ando and Uffe Haagerup and Cyril Houdayer and Amine Marrakchi}, journal={arXiv: Operator Algebras}, year={2018} }

We investigate the structure of the relative bicentralizer algebra ${\rm B}(N \subset M, \varphi)$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type ${\rm III_1}$ subfactor and $\varphi \in N_*$ is a faithful state. We first construct a canonical flow $\beta^\varphi : \mathbf R^*_+ \curvearrowright {\rm B}(N \subset M, \varphi)$ on the relative bicentralizer algebra and we show that the W$^*$-dynamical system $({\rm B}(N \subset M, \varphi), \beta^\varphi)$ is… Expand

#### 9 Citations

Full factors, bicentralizer flow and approximately inner automorphisms

- Mathematics
- 2018

We show that a factor M is full if and only if the $$C^*$$ C ∗ -algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of… Expand

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

- Mathematics
- 2018

We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion… Expand

Existentially closed W*-probability spaces

- Mathematics
- 2021

We study several model-theoretic aspects of W-probability spaces, that is, σ-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W-spaces and… Expand

Tensor product decompositions and rigidity of full factors

- Mathematics
- 2019

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product… Expand

Ozawa's class 𝒮 for locally compact groups and unique prime factorization of group von Neumann algebras

- Mathematics
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2020

Abstract We study class 𝒮 for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a… Expand

Noncommutative ergodic theory of higher rank lattices

- Mathematics
- 2021

We survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of higher rank lattices. These results have several applications to… Expand

Unique prime factorization for infinite tensor product factors

- Mathematics
- Journal of Functional Analysis
- 2019

In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property,… Expand

Charmenability of higher rank arithmetic groups

- Mathematics
- 2021

We complete the study of characters on higher rank semisimple lattices initiated in [BH19, BBHP20], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary… Expand

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

- Mathematics
- 2020

Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by… Expand

#### References

SHOWING 1-10 OF 47 REFERENCES

Ultraproducts of von Neumann algebras

- Mathematics
- 2012

We study several notions of ultraproducts of von Neumann algebras from a unifying viewpoint. In particular, we show that for a sigma-finite von Neumann algebra $M$, the ultraproduct $M^{\omega}$… Expand

Unique prime factorization and bicentralizer problem for a class of type III factors

- Mathematics
- 2015

We show that whenever $m \geq 1$ and $M_1, \dots, M_m$ are nonamenable factors in a large class of von Neumann algebras that we call $\mathcal C_{(\text{AO})}$ and which contains all free Araki-Woods… Expand

Fullness and Connes' $\boldsymbol{\tau}$ invariant of type III tensor product factors

- Mathematics
- 2016

We show that the tensor product $M \mathbin{\overline{\otimes}} N$ of any two full factors $M$ and $N$ (possibly of type ${\rm III}$) is full and we compute Connes' invariant $\tau(M… Expand

On the Relative Dixmier Property for Inclusions of C*-Algebras☆

- Mathematics
- 2000

Let N/M be an inclusion of von Neumann algebras with a conditional expectation E: M N satisfying the finite index condition of [PiPo], i.e., there exists c>0 such that E(x) cx, \x # M+ . In [Po4] we… Expand

Conne’s bicentralizer problem and uniqueness of the injective factor of type III1

- Mathematics
- 1987

In Connes' fundamental work "Classification of injective factors" [7], it is proved that injective factors of type III,t, 2 . 1 on a separable Hilbert space are completely classified by their "smooth… Expand

Singular MASAs in type III factors and Connes' Bicentralizer Property

- Mathematics
- 2017

We show that any type ${\rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal… Expand

Cohomology and Extensions of von Neumann Algebras. II

- Mathematics
- 1980

We develop a theory of extensions of von Neumann algebras by locally compact groups of automorphisms. The emphasis is on the description (from an algebraic point of view) of those extensions of a… Expand

Asymptotic structure of free product von Neumann algebras

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract Let (M, ϕ) = (M 1, ϕ1) * (M 2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra… Expand

Conditional Expectations in von Neumann Algebras

- Mathematics
- 1972

Let M be a von Neumann algebra and N its von Neumann subalgebra. Let ϑ be a faithful, semifinite, normal weight on M+ such that the restriction ϑ ¦ N of ϑ onto N is semifinite. The first main result… Expand

A Galois Correspondence for Compact Groups of Automorphisms of von Neumann Algebras with a Generalization to Kac Algebras

- Mathematics
- 1996

Abstract LetMbe a factor with separable predual andGa compact group of automorphisms ofMwhose action is minimal, i.e.,MG′∩M=C, whereMGdenotes theG-fixed point subalgebra. Then every intermediate von… Expand