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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. 

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 introduce a new iterative amalgamated free product construction of II factors, and use it to construct a separable II factor which does not have property Gamma and is not elementarily equivalent to the free group factor $$L(F_n)$$, for any $$n\geq 2$$. This provides the first explicit example of two non-elementarily equivalent $$II_1$$ factors without property Gamma. Moreover, our construction also provides the first explicit example of a $$II_1$$ factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory. 

We introduce a new class of groups called {\it wreath-like products}. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group $$G$$ in a natural family of wreath-like products with property (T) is W$^*$-superrigid: the group von Neumann algebra $$\text{L}(G)$$ remembers the isomorphism class of $$G$$. This allows us to provide the first examples (in fact, $$2^{\aleph_0}$$ pairwise non-isomorphic examples) of W$^*$-superrigid groups with property (T).