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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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A Random Matrix Approach to Absorption in Free Products
Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $$1$$-bounded entropy. We show that if $${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$$, then $${\mathcal{P}}$$ absorbs any subalgebra of $${\mathcal{M}}$$ that intersects it diffusely and that has $$1$$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $${\mathcal{P}} \leq{\mathcal{M}}$$ to have this absorption property, it suffices for $${\mathcal{M}}$$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto $${\mathcal{P}}$$.
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- PAR ID:
- 10290473
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2021
- Issue:
- 3
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 1919 to 1979
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa191
