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 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 $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite 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 e-prints
- Pub Date:
- June 2012
- arXiv:
- arXiv:1207.0054
- Bibcode:
- 2012arXiv1207.0054I
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Dynamical Systems;
- Mathematics - Group Theory;
- 46L36;
- 28D15;
- 37A20
- E-Print:
- This new (and final) version of the paper includes an appendix written jointly with Stefaan Vaes which fixes a gap from the previous version