Normalizers inside amalgamated free product von Neumann algebras
Summary: Recently, \textit{A. Ioana} [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 1, 71--130 (2015; Zbl 1351.46058)] 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 [loc. cit.] deduced this result from a more general dichotomy theorem on the normalizer \(\mathcal N_M(A)^{''}\) 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 \(L^\infty(X) \rtimes (\Gamma_1*\Gamma_2)\).
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