Asymptotic freeness in tracial ultraproducts

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 $u_1\in \mathcal{U}(M_1)$, $u_2\in \mathcal{U}(M_2)$ are Haar unitaries, the relative commutants $\{u_1{\}}^{\prime }\cap M^{\mathcal{U}}$ and $\{u_2{\}}^{\prime }\cap M^{\mathcal{U}}$ are freely independent in the ultraproduct $M^{\mathcal{U}}$. Our proof relies on Mei-Ricard's results [MR16] regarding $L^p$-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 $\mathrm{I}{\mathrm{I}}_1$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.

27 pages

References in corpus (3)

• Gamma stability in free product von Neumann algebras
• Strong convergence of tensor products of independent G.U.E. matrices
• Norm of matrix-valued polynomials in random unitaries and permutations

Cited by in corpus (1)

• Sequential commutation in tracial von Neumann algebras