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

@article{Ioana2012CartanSO,
title={Cartan subalgebras of amalgamated free product II\$\_1\$ factors},
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…
58 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
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
• Mathematics
Compositio 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
• Mathematics
Kyoto 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
• Mathematics