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Abstract We prove that ifAis a non-separable abelian tracial von Neuman algebra then its free powersA^∗n,2≤n≤∞, are mutually non-isomorphic and with trivial fundamental group,$$\mathcal{F}(A^{*n})=1$$, whenever 2≤n<∞. This settles the non-separable version of the free group factor problem. 

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. 

We prove that any product of two non-abelian free groups,\Gamma=\mathbb{F}_{m}\times\mathbb{F}_{k}, form,k\geq 2, is not Hilbert–Schmidt stable. This means that there exist asymptotic representations\pi_{n}\colon \Gamma\rightarrow \mathrm{U}({d_n})with respect to the normalized Hilbert–Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matricesA,Bsuch thatAalmost commutes withBandB^{*}, with respect to the normalized Hilbert–Schmidt norm, butA,Bare not close to any matricesA',B'such thatA'commutes withB'andB'^{*}. This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969.