Cartan subalgebras of amalgamated free product II$_1$ factors
Abstract
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and nonexistence 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 $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two nonhyperfinite countable ergodic probability measure preserving equivalence relations, then the II$_1$ factor $L(\mathcal R)$ has a unique Cartan subalgebra, up to unitary conjugacy. Finally, we show that the free product $M=M_1*M_2$ of any two II$_1$ factors does not have a Cartan subalgebra. More generally, we prove that if $A\subset M$ is a diffuse amenable von Neumann subalgebra and $P\subset M$ denotes the algebra generated by its normalizer, then either $P$ is amenable, or a corner of $P$ embeds into $M_1$ or $M_2$.
 Publication:

arXiv eprints
 Pub Date:
 June 2012
 arXiv:
 arXiv:1207.0054
 Bibcode:
 2012arXiv1207.0054I
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 46L36;
 28D15;
 37A20
 EPrint:
 This new (and final) version of the paper includes an appendix written jointly with Stefaan Vaes which fixes a gap from the previous version