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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$$\mathrm {II_1}$$factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.
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This content will become publicly available on April 1, 2026
Internal Sequential Commutation and Single Generation
Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof.
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- Award ID(s):
- 2350049
- PAR ID:
- 10595208
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 8
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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