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

TL;DR

This paper proves the absence of Cartan subalgebras in nontracial free product von Neumann algebras and establishes unique Cartan decomposition results for certain group measure space factors arising from amalgamated free product groups.

Contribution

It generalizes the nonexistence of Cartan subalgebras to nontracial free product von Neumann algebras and proves unique Cartan decomposition for specific group measure space factors.

Findings

Nonamenable free product von Neumann algebras lack Cartan subalgebras.

Countable ergodic equivalence relations split as free products have associated factors with unique Cartan subalgebras.

Group measure space factors from amalgamated free product groups have unique Cartan decomposition.

Abstract

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, \varphi_1) \ast (M_2, \varphi_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined on a standard measure space and which splits as the free product $\mathcal R = \mathcal R_1 \ast \mathcal R_2$ of recurrent subequivalence relations gives rise to a nonamenable factor $\rL(\mathcal R)$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors $\rL^\infty(X) \rtimes \Gamma$ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu)$ on standard measure spaces of amalgamated groups $\Gamma = \Gamma_1 \ast_{\Sigma} \Gamma_2$ over a finite subgroup $\Sigma$.