Publications of the Research Institute for Mathematical Sciences

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Volume 50, Issue 4, 2014, pp. 695–721
DOI: 10.4171/PRIMS/147

Published online: 2014-10-22

Normalizers inside Amalgamated Free Product von Neumann Algebras

Stefaan Vaes[1]

(1) Katholieke Universiteit Leuven, Belgium

Recently, Adrian Ioana proved that all crossed products $L^\infty(X) \rtimes (\Gamma_1*\Gamma_2)$ by free ergodic probability measure preserving actions of a nontrivial free product group $\Gamma_1 * \Gamma_2$ have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer $\mathcal N_M(A)^{\prime\prime}$ of an amenable subalgebra $A$ of an amalgamated free product von Neumann algebra $M = M_1 *_B M_2$. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in $L^\infty(X) \rtimes (\Gamma_1*\Gamma_2)$.

Keywords: amalgamated free products, Cartan subalgebras, deformation/rigidity theory, nonsingular group actions

Vaes Stefaan: Normalizers inside Amalgamated Free Product von Neumann Algebras. Publ. Res. Inst. Math. Sci. 50 (2014), 695-721. doi: 10.4171/PRIMS/147