Adrian Ioana, Sorin Popa, Stefaan Vaes
We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra L(G) completely "remembers" the group G. More precisely, if L(G) is isomorphic to the von Neumann algebra L(Λ) of an arbitrary countable group Λ, then Λ must be isomorphic to G. This represents the first superrigidity result pertaining to group von Neumann algebras.
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We prove that for many nonamenable groups Γ, including all hyperbolic groups and all nontrivial free products, the left-right wreath product group G := (Z/2Z)^(Γ) \rtimes (Γ\times Γ) is W*-superrigid. This means that the group von Neumann algebra LG entirely remembers G. More precisely, if LG is isomorphic with LΛfor an arbitrary countable group Λ, then Λmust be isomorphic with G.
W*-superrigidity for cocycle twisted group von Neumann algebras
本文构造可数群，证明其群 von Neumann 代数即使带任意 2-上循环扭曲，双有限双模意义下的虚拟同构仍强制底层群虚拟同构，并给出不可分解的 II_1 型因子。
Rigidity for von Neumann algebras of graph product groups II. Superrigidity results
本文证明了花状图积群Γ可从任意无穷顶点群的非平凡图积群生成的冯·诺伊曼代数范畴中被完全识别，获得了更精确的C*-代数版本，并将W*-超刚性结论推广至性质(T) wreath-like product群的直积。
McDuff superrigidity for group II$_1$ factors
该研究发现了首个ICC群G，其冯·诺依曼代数为麦迪夫，并揭示了新的超刚性现象，即满足同构的群可分解为原群与可递群的乘积。
W*-superrigidity for groups with infinite center
本文构造了具有无限中心但仍具 W*-超刚性的离散群，其 von Neumann 代数完全决定群结构，方法结合商群与 2-上同调的刚性。
A group $G$ is called $W^*$-superrigid (resp. $C^*$-superrigid) if it is completely recognizable from its von Neumann algebra $L(G)$ (resp. reduced $C^*$-algebra $C_r^*(G)$). Developing new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of $W^*$-superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in $C^*$-algebra theory including additional examples of $C^*$-superrigid groups and explicit computations o…
Non-amenable C$^*$-superrigid groups that are not W$^*$-superrigid
本文构造了非amenable且无法从群von Neumann代数重建的C$^*$-超刚群，并展示了同时满足C$^*$-和W$^*$-超刚性质的自由积群例子。
Wreath-like products of groups and their von Neumann algebras I: $W^\ast$-superrigidity
本文定义类圈积群，证明一类具有 Kazhdan 性质 (T) 的类圈积可由其群 von Neumann 代数唯一恢复同构类，并给出 $2^{\aleph_0}$ 个此类 $W^*$-超刚性群。
In this paper we explore a generic notion of superrigidity for von Neumann algebras $L(G)$ and reduced $C^*$-algebras $C^*_r(G)$ associated with countable discrete groups $G$. This allows us to classify these algebras for various new classes of groups $G$ from the realm of coinduced groups.
We show that if G is a discrete group which does not have the Haagerup property but does have an unbounded cocycle into a C_0 representation and if G acts on a finite von Neumann algebra B such that the inclusion B \subset (B \rtimes G) has the Haagerup property from below then any group-measure space Cartan subalgebra must have a corner which embeds into B inside B \rtimes G. Taking the action to be trivial we produce examples of II_1 factors N such that N \otimes M is not a group-measure space construction whenever M is a finite factor with the Haagerup property. Taking the action on a proba…
W*-superrigidity for discrete quantum groups
本文研究了离散量子群的W*-超rigidity性质，证明了某些自然的co-induced离散量子群具有此性质，同时指出大多数现有W*-超rigid群不满足量子W*-超rigidity。
Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups CR satisfying the following property: given any groups G_1, G_2 in CR, then any free, ergodic, measure preserving action on a probability space G_1 x G_2 on X gives rise to a von Neumann algebra with unique group measure space Cartan subalgebra. Pairing this result with Popa's Orbit Equivalence Superrigidity Theorem we obtain new examples of W*-superrigid actions.
We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W$^*$-superrigid) actions that can be completely recovered from their von Neumann algebras and II$_1$ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.
We show that several important normal subgroups $Γ$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $Γ\curvearrowright X$ is stably OE-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preservi…
For $n\geq 3,$ let $Γ=SL_n(\mathbb Z).$ We prove the following superridigity result for $Γ$ in the context of operator algebras. Let $L(Γ)$ be the von Neumann algebra generated by the left regular representation of $Γ.$ Let $M$ be a finite factor and let $U(M)$ be its unitary group. Let $π: Γ\to U(M)$ be a group homomorphism such that $π(Γ)''=M.$ Then \begin{itemize} \item[(i)] either $M$ is finite dimensional, or \item [(ii)] there exists a subgroup of finite index $Λ$ of $Γ$ such that $π|_Λ$ extends to a homomorphism $U(L(Λ))\to U(M).$ \end{itemize} The result is deduced from a complete desc…
We generalize W*-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If Γ is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action σ of Γ is W*-superrigid. More precisely, if ρ is another free ergodic action of a group Λ such that the crossed-product von Neumann algebras associated with ρ and σ are isomorphic, then Λ and Γ are isomorphic, and the actions ρ and σ are conjugate. We prove a similar statement whenever Γ is a non-amenable…
ICC property (T) groups without W$^*$-superrigidity
该研究构造了两个不同构但群冯·诺依曼代数同构的ICC性质(T)群，为ICC性质(T)群的Connes刚性猜想提供了反例，结果借助GPT-5.6 Sol完成。
We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with uncountable fundamental groups and the construction of W*-superrigid actions where the crossed product entirely remembers the initial group action that it was constructed from.
Von Neumann Orbit Equivalence
本文提出冯·诺伊曼轨道等价的概念，证明其在自由积和图积下的稳定性，并探讨了群冯·诺伊曼代数的等价关系。
In this short note we classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces.
The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung in arXiv:math/0510576 . This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all Property (T) von Neumann algebras with finite dimensional center and group von Neumann algebras of Property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-…
We study qualitative properties of the group von Neumann algebra of a Baumslag-Solitar group. Namely, we prove that, in the non-amenable and {ICC} case, the associated ${\rm II}_1$ factor is prime, not solid, and does not have any Cartan subalgebra.
We construct inner amenable groups G with infinite conjugacy classes and such that the associated II_1 factor does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.
Reduced HNN extensions of von Neumann algebras (as well as $C^*$-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.
We prove that if a countable discrete group $Γ$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $Γ= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $Γ= H \times H'$ with $H$ an infinite Kazhdan group and $H'$ arbitrary), and $\Cal V$ is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. $\Cal V$ countable discrete, or separable compact), then any $\Cal V$-valued measurable cocycle for a measure preserving action $Γ\curvearrowright X$ of $Γ$ on a probability space $(X,μ)$ which is weak mixing on $H$ and {\it s-malleable} …
We first show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. We then address co-amenability for von Neumann algebras, describing notably how it relates to the former.
In this talk, I will survey recent progress made on the classification of von Neumann algebras arising from countable groups and their actions on probability spaces. In particular, I will present the first results which provide classes of (superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras.
We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.
We introduce "embedding dimensions" of a family of generators of a finite von Neumann algebra when the von Neumann algebra can be faithfully embedded into the ultrapower of the hyperfinite II$_1$ factor. These embedding dimensions are von Neumann algebra invariants, i.e., do not depend on the choices of the generators. We also find values of these invariants for some specific von Neumann algebras.
In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let N and M be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner-Popa. If N and M are sufficiently close, then N and M are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.
We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any non-injective subfactor in a hyperbolic group von Neumann algebra is non-Gamma and prime. The proof is based on C*-algebra theory.
The paper is devoted to the description of $2$-local derivations on von Neumann algebras. Earlier it was proved that every $2$-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type $III$ von Neumann algebras. This implies that on arbitrary von Neumann algebra each $2$-local derivation is a derivation.
In this paper we describe the amalgamated free product of finite and semifinite hyperfinite von Neumann algebras over atomic type I subalgebras. To do this we extend the notions of free dimension and standard embeddings used in the related results for finite von Neumann algebras to ones which work better for the semifinite case. We also define classes R3 (of finite von Neumann algebras) and R4 (of semifinite von Neumann algebras) which are closed under such amalgamated free products.
Weak exactness and amalgamated free product of von Neumann algebras
本文证明弱正合冯·诺依曼代数的融合自由积仍弱正合，方法利用Toeplitz-Pimsner代数的泛性质及双模上的局部凸拓扑刻画弱正合性。
We give a complete description of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II$_1.$
Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or $W^*$-objects among the Hilbert ($C^*$-)modules are around since the first papers by Paschke (1973) and Rieffel (1974) that lift Kaplansky's setting (1953) to modules over noncommutative $C^*$-algebras. While the formal definition of $W^*$-modules} is due to Baillet, Denizeau, and Havet (1988), the one of von Neumann modules as strongly closed operator spaces started with Skeide (2000). It has been paired with the definition…
The Stone spectrum of a von Neumann algebra is a generalization of the Gelfand spectrum, as was shown by de Groote. In this article we clarify the structure of the Stone spectra of von Neumann algebras of type $I_{n}$.
In the present paper we prove that every 2-local derivation on a semi-finite von Neumann algebra is a derivation.
Types of graded tensor products of graded von Neumann algebras
本文针对分级冯·诺依曼代数，在特殊情形下确定了其分级张量积的类型，并完善了相应的乘法表。
In the present paper we prove that every 2-local derivation on a von Neumann algebra of type I is a derivation.
In this paper, we prove a non-commutative version of the Weyl-von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous ${\rm C}^{\ast}$-algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors.
In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.
We obtain a characterization of property (T) for von Neumann algebras in terms of 1-cohomology similar to the Delorme-Guichardet Theorem for groups.
In this paper we describe the amalgamated free product of two hyperfinite von Neumann algebras over a finite dimensional subalgebra. In general the free product is a finite direct sum of interpolated free group factors and a hyperfinite von Neumann algebra. We then show that the class of von Neumann algebras of this form is closed under taking amalgamated free products over finite dimensional subalgebras.
We consider cones in a Hilbert space associated to two von Neumann algebras and determine when one algebra is included in the other. If a cone is assocated to a von Neumann algebra, the Jordan structure is naturally recovered from it and we can characterize projections of the given von Neumann algebra with the structure in some special situations.
In this paper, we discuss some equivalent definitions of Property $Γ$ for a type II$_1$ von Neumann algebra. Using these equivalent definitions, we prove that the Pisier's similarity degree of a type II$_1$ von Neumann algebra with Property $Γ$ is equal to $3$.
We apply the ideas of Muhly, Skeide and Solel [MSS03] of considering von Neumann B-modules as intertwiner spaces for representations of B' to obtain a new, simple and self-contained proof for self-duality of von Neumann modules. This simplifies also the approach of [MSS03].
Using Speicher central limit Theorem we provide Hiai's q-Araki-Woods von Neumann algebras with nice asymptotic matricial models. Then, we use this model and an elaborated ultraproduct procedure, to show that all q-Araki-Woods von Neumann algebras are QWEP.
We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm{I}\sb{\infty}.$
It is shown that if a bipartite behavior admits a field representation in which Alice (or Bob's) observable algebra generates a purely atomic von Neumann algebra then it is non-relativistic.
We give a positive answer, in the measurable-group-theory context, to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors.
A theory of Galois co-objects for von Neumann bialgebras is introduced. This concept is closely related to the notion of comonoidal W*-Morita equivalence between von Neumann bialgebras, which is a Morita equivalence taking the comultiplication structure into account. We show that the property of `being a von Neumann algebraic quantum group' (i.e. `having invariant weights') is preserved under this equivalence relation. We also introduce the notion of a projective corepresentation for a von Neumann bialgebra, and show how it leads to a construction method for Galois co-objects and comonoidal W*…
We prove that every surjective isometry between the unit spheres of two von Neumann algebras admits a unique extension to a surjective real linear isometry between these two algebras.
We define $Γ_q(B,S \otimes H)$, the generalized $q$-gaussian von Neumann algebras associated to a sequence of symmetric independent copies $(π_j,B,A,D)$ and to a subset $1 \in S = S^* \subset A$ and, under certain assumptions, prove their strong solidity relative to $B$. We provide many examples of strongly solid generalized $q$-gaussian von Neumann algebras. We also obtain non-isomorphism and non-embedability results about some of these von Neumann algebras.
We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras.
We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.
We show that the second cohomology group $H^2(M\overline\otimes N, M\overline\otimes N)$ is always zero for arbitrary type II$_1$ von Neumann algebras $M$ and $N$.
Based on the notion of free orbit-dimension introduced by D. Hadwin and J. Shen [4], we introduce a new invariant on finite von Neumann algebras that do not necessarily act on separable Hilbert space. We show that this invariant is independent on the generating set, and we extend some results in [4] to von Neumann algebras that are not finitely generated.
Strongly facially symmetric spaces isometrically isomorphic to the predual space of an atomic commutative von Neumann algebra are described
We study certain $q$-deformed analogues of the maximal abelian subalgebras of the group von Neumann algebras of free groups. The radial subalgebra is defined for Hecke deformed von Neumann algebras of the Coxeter group $(\mathbb{Z}/{2\mathbb{Z}})^{\star k}$ and shown to be a maximal abelian subalgebra which is singular and with Pukánszky invariant $\{\infty\}$. Further all non-equal generator masas in the $q$-deformed Gaussian von Neumann algebras are shown to be mutually non-unitarily conjugate.
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