A. We show that if Γ = Γ 1 × · · · × Γn is a product of n ≥ 2 non-elementary ICC hyperbolic groups then any discrete group Λ which is W * -equivalent to Γ decomposes as a k-fold direct sum exactly when k = n. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group Λ. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II 1 factors.

In this paper we introduce a new family of icc groups Γ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II 1 factor L(Γ) arise only from the canonical direct product decompositions of the underlying group Γ. Our groups are assembled from certain HNNextensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence we obtain several new examples of groups that give rise to prime factors.of the corresponding II 1 factor L(Γ) can arise only from the canonical direct product decompositions of the underlying group Γ. Pant and the second author showed the same result holds when Γ is a poly-hyperbolic group with non-amenable factors in its composition series [dSP17]. In this paper we introduce several new classes of groups for which this tensor product rigidity phenomenon still holds. Our groups arise from natural algebraic constructions involving HNN-extensions or amalgamated free products thus including many remarkable groups intensively studied throughout mathematics such as graph products or poly-amalgams groups.Basic properties in Bass-Serre theory of groups show that the only way an amalgam Γ 1 * Σ Γ 2 could decompose as a direct product is through its core Σ.An interesting question is to investigate situations when this basic group theoretic aspect could be upgraded to the von Neumann algebraic setting. It is known this fails in general since there are examples of product indecomposable icc amalgams whose corresponding factors are McDuff and hence decomposable as tensor products (see Remarks 6.3). However, under certain indecomposability assumptions on the core algebra (see also Remarks 6.3) we are able to provide a positive answer to our question.Theorem A. Let Γ = Γ 1 * Σ Γ 2 be an icc group such that [Γ 1 : Σ] ≥ 2 and [Γ 2 : Σ] ≥ 3. Assume that Σ is finite-by-icc and any corner of L(Σ) is virtually prime. Suppose that L2 , for some groups Σ 0 Γ 0 1 , Γ 0 2 , and hence Γ = Ω × (Γ 0 1 * Σ 0 Γ 0 2 ). Moreover, there is a unitary u ∈ L(Γ), t > 0, and a permutation s of {1, 2} such that M s(1) = uL(Ω) t u * and M s(2) = uL(Γ 0 1 * Σ 0 Γ 0 2 ) 1/t u * .(1.1)The same question can be considered in the realm of non-degenerate HNN-extension groups and a similar approach leads to the following counterpart of Theorem A. Theorem B. Let Γ = HNN(Λ, Σ, θ) be an icc group such that Σ = Λ = θ(Σ). Assume that Σ is finite-by-icc and any corner of L(Σ) is virtually prime. Suppose that L(Γ) = M 1⊗ M 2 , for diffuse M i 's. Then there exist decompositions Σ = Ω × Σ 0 with Σ 0 finite and Λ = Ω × Λ 0 . In addition, there is ω ∈ Ω such that θ = ad(ω) |Ω × θ |Σ 0 : Ω × Σ 0 → Ω × Λ 0 and hence Γ = Ω × HNN(Λ 0 , Σ 0 , θ |Σ 0 ). Also, there is a unitary u ∈ L(Γ), t > 0, and a permutation s of {1, 2} such that M s(1) = uL(Ω) t u * and M s(2) = uL(HNN(Λ 0 , ...

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IntroductionIn the early years of the twentieth century, the foundations for quantum mechanics were laid out by Dirac, Heisenberg, Bohr, Schrödinger, and others. In his work on the foundations of quantum mechanics, John von Neumann postulated that physical phenomena should be modeled in terms of Hilbert spaces and operators, with observables corresponding to self-adjoint operators and states corresponding to unit vectors. Motivated by his interest in the theory of single operators, he would introduce the notion of what is now termed a von Neumann algebra. Von Neumann and Francis Murray subsequently published a series of fundamental papers, beginning with "On rings of operators" [13], that develop the basic properties of these algebras and establish operator algebras as an independent field of study.In the years after Murray and von Neumann's initial work, the field of operator algebras developed rapidly and split into subfields including * -algebras and von Neumann algebras. Moreover, operator algebraists began to examine generalizations of these objects, such as operator spaces and operator systems. The importance of operator algebras can be witnessed by its applications in Voiculescu's free probability theory, Popa's deformation/rigidity theory, and Jones' theory of subfactors. These areas give us insight into numerous fields, including random matrix theory, quantum field theory, ergodic theory, and knot theory.In a landmark paper unraveling the isomorphism classes of injective von Neumann algebras, Connes proves that it is possible to construct a sequence of approximate embeddings for a large class of von Neumann algebras into finite-dimensional matrix algebras; Connes somewhat casually remarks that this property should hold for all separable von Neumann algebras. Formally, Connes' embedding problem, as this assertion is now called, asks if every type II 1 factor acting on a separable Hilbert space is embeddable into an ultrapower of the hyperfinite II 1 factor via a nonprinciple ultrafilter.Our goal is to unravel the meaning behind Connes' embedding problem and to highlight its significance by providing equivalent formulations that have driven research in the field.

This work is a compilation of structural results for the von Neumann algebras of poly-hyperbolic groups established in a series of works done jointly with I. Chifan and T. Sinclair; and S. Pant. These works provide a wide range of circumstances where the product structure, a discrete structural property, can be recovered from the von Neumann algebra (a continuous object). The primary result of Chifan, Sinclair and myself is as follows: if Γ = Γ 1 × • • • × Γ n is a product of non-elementary hyperbolic icc groups and Λ is a group such that L(Γ) ∼ = L(Λ), then Λ decomposes as an n-fold product of infinite groups. This provides a group-level strengthening of the unique prime decomposition of Ozawa and Popa by eliminating any assumption assumption on the target group Λ. The methods necessary to establish this result provide a malleable procedure which allows one to rebuild the product of a group from the algebra itself. Modifying the techniques found in the previous work, Pant and I are able to demonstrate that the class of poly-groups exhibit a similar phenomenon. Specifically, if Γ is a poly-hyperbolic group whose corresponding algebra is non-prime, then the group must necessarily decompose as a product of infinite groups.