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- W4387156635 abstract "We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 ast M_2$ is a tracial free product von Neumann algebra and $u_1 in mathscr U(M_1)$, $u_2 in mathscr U(M_2)$ are Haar unitaries, the relative commutants ${u_1}' cap M^{mathcal U}$ and ${u_2}' cap M^{mathcal U}$ are freely independent in the ultraproduct $M^{mathcal U}$. Our proof relies on Mei-Ricard's results [MR16] regarding $operatorname{L}^p$-boundedness (for all $1 < p < +infty$) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan-Ioana-Kunnawalkam Elayavalli's recent construction [CIKE22] to provide the first example of a ${rm II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras." @default.
- W4387156635 created "2023-09-30" @default.
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- W4387156635 date "2023-09-26" @default.
- W4387156635 modified "2023-09-30" @default.
- W4387156635 title "Asymptotic freeness in tracial ultraproducts" @default.
- W4387156635 doi "https://doi.org/10.48550/arxiv.2309.15029" @default.
- W4387156635 hasPublicationYear "2023" @default.
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