# Cartan subalgebras of amalgamated free product II$_1$ factors

@article{Ioana2012CartanSO, title={Cartan subalgebras of amalgamated free product II\$\_1\$ factors}, author={Adrian Ioana}, journal={arXiv: Operator Algebras}, year={2012} }

We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $\Gamma$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ arising from a free ergodic probability measure preserving action of $\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if…

## 70 Citations

### Unbounded derivations, free dilations and indecomposability results for II$_1$ factors

- Mathematics
- 2012

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra.…

### On the vanishing cohomology problem for cocycle actions of groups on II_1 factors

- MathematicsAnnales Scientifiques de l'École Normale Supérieure
- 2021

We prove that any free cocycle action of a countable amenable group $\Gamma$ on any II$_1$ factor $N$ can be perturbed by inner automorphisms to a genuine action. This {\em vanishing cohomology}…

### Amalgamated free product type III factors with at most one Cartan subalgebra

- MathematicsCompositio Mathematica
- 2013

Abstract We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$.…

### Thin II1 factors with no Cartan subalgebras

- MathematicsKyoto Journal of Mathematics
- 2019

It is a wide open problem to give an intrinsic criterion for a II_1 factor $M$ to admit a Cartan subalgebra $A$. When $A \subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is "simple" in…

### Factorial relative commutants and the generalized Jung property for II1 factors

- MathematicsAdvances in Mathematics
- 2021

### Unitary conjugacy for type III subfactors and W$^*$-superrigidity

- MathematicsJournal of the European Mathematical Society
- 2021

Let $A,B\subset M$ be inclusions of $\sigma$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa's…

### Prime II1 factors arising from actions of product groups

- MathematicsJournal of Functional Analysis
- 2020

### Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras

- MathematicsAnalysis & PDE
- 2020

For a right-angled Coxeter system $(W,S)$ and $q>0$, let $\mathcal{M}_q$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators $T_s, s \in S$ satisfying the Hecke…

### Asymptotic structure of free Araki–Woods factors

- Mathematics
- 2014

The purpose of this paper is to investigate the structure of Shlyakhtenko’s free Araki–Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free…

### Non-classification of Cartan subalgebras for a class of von Neumann algebras

- MathematicsAdvances in Mathematics
- 2018

## References

SHOWING 1-10 OF 65 REFERENCES

### Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups

- Mathematics
- 2014

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$Fn↷(X,μ) of a free group on n generators $${\mathbb{F}_n, 2\leq n \leq…

### Uniqueness of the group measure space decomposition for Popa's $\Cal H\Cal T$ factors

- Mathematics
- 2011

We prove that every group measure space II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ coming from a free ergodic rigid (in the sense of [Po01]) probability measure preserving action of a group $\Gamma$…

### Free products, Orbit Equivalence and Measure Equivalence Rigidity

- Mathematics
- 2008

We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely…

### Group measure space decomposition of II1 factors and W*-superrigidity

- Mathematics
- 2010

We prove a “unique crossed product decomposition” result for group measure space II1 factors L ∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ…

### Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I

- Mathematics
- 2006

We consider crossed product II1 factors $M = N\rtimes_{\sigma}G$, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G…

### A class of groups for which every action is W$^*$-superrigid

- Mathematics
- 2010

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A \rtimes \Gamma covering certain cases where \Gamma is an amalgamated free product over a non-amenable…

### Bass-Serre rigidity results in von Neumann algebras

- Mathematics
- 2008

We obtain new Bass-Serre type rigidity results for ${\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard…

### Mixing subalgebras of finite von Neumann algebras

- Mathematics
- 2009

Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties…

### Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

- Mathematics
- 2005

We consider amalgamated free product II1 factors M = M1*BM2*B… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be…

### Strongly 1-Bounded Von Neumann Algebras

- Mathematics
- 2005

Abstract.Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if
$${\mathbb{P}}^\alpha (F) < \infty$$
where
$${\mathbb{P}}^\alpha$$
is…