Normalizers inside Amalgamated Free Product von Neumann Algebras

  • Stefaan Vaes

    Katholieke Universiteit Leuven, Belgium


Recently, Adrian Ioana proved that all crossed products L(X)(Γ1Γ2)L^\infty(X) \rtimes (\Gamma_1*\Gamma_2) by free ergodic probability measure preserving actions of a nontrivial free product group Γ1Γ2\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 NM(A)\mathcal N_M(A)^{\prime\prime} of an amenable subalgebra AA of an amalgamated free product von Neumann algebra M=M1BM2M = 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(X)(Γ1Γ2)L^\infty(X) \rtimes (\Gamma_1*\Gamma_2).

Cite this article

Stefaan Vaes, Normalizers inside Amalgamated Free Product von Neumann Algebras. Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, pp. 695–721

DOI 10.4171/PRIMS/147