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}
}
  • A. Ioana
  • Published 30 June 2012
  • Mathematics
  • arXiv: Operator Algebras
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… 

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References

SHOWING 1-10 OF 65 REFERENCES

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

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

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

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

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

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

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

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

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

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

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
...