Sorin Popa
We give a new proof of a result of Ozawa showing that if a von Neumann subalgebra $Q$ of a free group factor $L\Bbb F_n, 2\leq n\leq \infty$ has relative commutant diffuse (i.e. without atoms), then $Q$ is amenable.
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We explicit Ozawa kernels for classical group theoretical constructions, for discrete metric spaces of finite asymptotic dimension, of large enough Hilbert space compression, and for suitable actions of countable groups on metric spaces. We also obtain an alternative proof of stability results concerning Yu's property A.
We show that the group factors of ICC lattices in either SO(n,1) or SU(n,1), n \geq 2, are strongly solid in the sense of Ozawa and Popa. This strengthens a result of Ozawa and Popa showing that these factors do not have Cartan subalgebras.
We prove a Kurosh type theorem for free-product type II_1 factors. In particular, if M = LF_2 \otimes R, then the free-product type II_1 factors M*...*M are all prime and pairwise non-isomorphic. This paper is a continuation of [N. Ozawa, Solid von Neumann algebras] and [N. Ozawa and S. Popa, Some prime factorization results for type II_1 factors].
We study class $\mathcal S$ for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class $\mathcal S$ and prove unique prime factorization results for group von Neumann algebras of products of locally compact groups in this class. We also prove that class $\mathcal S$ is a measure equivalence invariant.
We consider von Neumann algebras generated by the stationary laws of free stochastic differential equations of the form $dX_t = dS_t -1/2 DV(X_t)$ for a suitably convex multivariate noncommutative polynomial $V$. Using techniques of Guionnet and Shlyakhtenko, we prove that these von Neumann algebras have property AO of Ozawa and are thus solid.
Ozawa showed that for any i.c.c., hyperbolic group, the associated group factor is solid. Developing a new approach that combines some methods of Peterson, Ozawa and Popa, and Ozawa, we strengthen this result by showing that these factors are strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana, we show that profinite actions of lattices in Sp(n,1), n>1, are virtually W*-superrigid.
We characterize when the reduced C*-algebra of a group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C*-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C*-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.
We show that Ozawa's recent results on solid von Neumann algebras imply that there are free Araki-Woods factors, which fail to have free absorption. We also show that a free Araki-Woods factors $Γ(μ, n)$ associated to a measure and a multiplicity function $n$ may non-trivially depend on the multiplicity function.
We prove a generalization of N. Ozawa's Kurosh-type theorem to the setting of free products of semiexact II_1 factors with respect to arbitrary (non-tracial) faithful normal states. We are thus able to distinguish certain resulting type III factors. For example, if M = LF_n \otimes LF_m and {ϕ_i} is any sequence of faithful normal states on M, then the l-various (M,ϕ_1) * ... * (M,ϕ_l) are all mutually non-isomorphic.
We consider some conditions similar to Ozawa's condition (AO), and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a corollary, we prove that $\rm II_1$ factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type $\rm III_1$ factors with such a condition are semisolid as a $\rm II_\infty$ factor.
We prove some unique factorization results for tensor products of free quantum group factors. They are type III analogues of factorization results for direct products of bi-exact groups established by Ozawa and Popa. In the proof, we first take continuous cores of the tensor products, which satisfy a condition similar to condition (AO), and discuss some factorization properties for the continuous cores. We then deduce factorization properties for the original type III factors. We also prove some unique factorization results for crossed product von Neumann algebras by direct products of bi-exac…
We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that the associated von Neumann algebras are full II_1-factors and Brannan's result showing that the orthogonal free quantum groups have Haagerup's approximation property. We also deduce Ozawa-Popa's property strong (HH) and give a new proof of Isono's result about strong solidity.
This paper includes a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors \cite{CS}, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance we show that every II$_1$ factor associated with a weakly amenable group in the class $\mathcal S$ of Ozawa is strongly solid, \cite{OzSolid}. There is also the following product version of this result: any maximal abelian $\star$-subal…
Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $σ$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenabi…
We introduce the notion of L^2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L^2-rigidity passes to normalizers and is satisfied by nonamenable II_1 factors which are non-prime, have property $Γ$, or are weakly rigid. As a consequence we obtain that if $M$ is a free product of diffuse von Neumann algebras, or if $M = LΓ$ where $Γ$ is a finitely generated group with $b_1^{(2)}(Γ) > 0$, then any nonamenable regular subfactor of $M$ is prime and …
Strongly solid ${\rm II_1}$ factors with an exotic MASA
本文通过扩展 Ozawa 和 Popa 的技术，构造了一个包含奇异极大交换子代数的非顺从强刚性 $\ m{II}_1$ 因子，证明了存在具有该性质但无 Cartan 子代数的因子，并基于自由熵结果阐明了其非插值自由群因子等性质。
Relative Biexactness for Relative Hyperbolic Groups and Some Applications
本文证实Ozawa等人关于有限生成相对双曲精确群为相对双精确的猜想，由此得到素群冯·诺依曼代数，并构造了两两非稳定*同构的群测度空间冯·诺依曼代数族。
PHP decompositions and primitivity of group rings for linear groups
本文证明具有小泽性质PHP且含非阿贝尔自由子群的可数群的群环对任意域本原，并推论非平凡可数线性群若具有平凡可和根则也满足此性质，同时建立了PHP类的一些永久性质。
We give an general estimate for the non-microstates free entropy dimension $δ^{*}(X_{1},..., X_{n})$. If $X_{1},..., X_{n}$ generate a diffuse von Neumann algebra, we prove that $δ^{*}(X_{1},..., X_{n})\geq 1$. In the case that $X_{1},..., X_{n}$ are $q$-semicircular variables as introduced by Bozejko and Speicher and $q^{2}n<1$, we show that $δ^{*}(X_{1},..., X_{n})>1$. We also show that for $|q|<\sqrt{2}-1$, the von Neumann algebras generated by a finite family of $q$-Gaussian random variables satisfy a condition of Ozawa and are therefore solid: the relative commutant of any diffuse subalge…
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…
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors
本文证明自由 Araki-Woods 因子具有完全度量逼近性质，并利用 Ozawa-Popa 技术证明特定非顺从子因子无 Cartan 子代数，最终解决了 Connes 构造的 III_1 型因子与自由 Araki-Woods 因子不同构的问题。
In this paper we provide new examples of property (T) group factors with trivial fundamental group thereby providing more evidence towards Popa's conjecture on triviality of fundamental groups for property (T) group factors (page 9 \cite{Po13}; see also Problem 2, page 551 in Connes' book \cite{Co94}). Our groups arise as direct product of groups either in Class $\mathscr S$ or Class $\mathscr V$, as introduced in \cite{CDHK20}. We establish a unique prime factorization result for these product groups, thereby providing more evidence towards Popa's conjecture on prime decomposition of property…
We show that if $Γ= Γ_1\times\dotsb\times Γ_n$ is a product of $n\geq 2$ 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.
We show a classification theorem for locally compact separable abelain group actions on factorsw with the Rohlin property. This is a generalization of Masuda and Tomatsu's work.
We study the matricial field (MF) property for certain reduced crossed product C*-algebras and their traces. Using classification techniques and induced K-theoretic dynamics, we show that reduced crossed products of ASH-algebras of real rank zero by free groups are MF if and only if they are stably finite. We also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties. Combining these results with recent progress in Elliott's Classification Program, it follows that if $A$ is a separable, simple, unital, nuclear, monotracial C*-al…
Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $τ$. We prove that if $Γ$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M\rtimesΓ$ has also the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed products von Neumann algebras are $\textrm{II}_1$-factors with the Haagerup proper…
On the (Local) Lifting Property
本文研究了（局部）提升性质（(L)LP），通过提升星-同态来刻画该性质，并证明了若A和B具有LP且F是它们的有限维C*-子代数，则A*F B具有LP。同时，证明了Exel的软 тор具有LP，并由此得出自由积的C*-代数是诱导极限的RFD C*-代数。此外，还证明了对于一类包含C*(F_n×F_n)的C*-代数，LLP等价于Ext为群。
Spectral gap characterizations of property (T) for II$_1$ factors
该研究针对II$_1$因子，证明了性质(T)与嵌入更大迹型冯·诺依曼代数的弱谱间隙等价，还通过弱混合双模的几乎中心向量性质得到仅需不可约嵌入弱谱间隙的更强刻画。
Complete Metric Approximation Property For Mixed $q$-Deformed Araki-Woods Factors
该研究利用超积嵌入方法，针对元素在-1到1之间的对称矩阵，证明了混合$q$-形变Araki-Woods因子具有弱*完全度量逼近性质，丰富了算子代数领域的相关理论。
We show that any type ${\rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type ${\rm III_1}$ factors.
Relative bi-exactness and structural results for graph-wreath product von Neumann algebras
本文研究了图- wreath积 von Neumann 代数的相对双完全性，通过 $C^{\ast}$-代数方法证明了其结构结果，并建立了稳定同构下的刚性结果及新的素 $\mathrm{II}_1$ 因子。
Consequences of the random matrix solution to the Peterson-Thom conjecture
本研究借助Peterson-Thom猜想的随机矩阵解，推导自由群因子的多项新结构性质，解决粗性猜想、推广强稳固性结果，还得出子代数缠结二分性及超积嵌入应用，推动算子代数领域相关研究。
Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions
本文证明了具有PHP性质的离散群生成的减少扭曲群C*代数是完全自洽的，并展示了某些有限由G扩张的群的减少扭曲C*代数具有稳定秩一且为纯，从而严格比较。
The main result of this paper is that the k^{\rm th} continuous Hochschild cohomology groups H^k(\cl M,\cl M) and H^k(\cl M,B(H)) of a von Neumann factor ${\cl M}\subseteq B(H) of type {\rm II}_1 with property Gamma are zero for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the $\|\cdot\|_2$-norm of separately ultraweakly continuous multilinear maps, and combine these results to reduce to the case of completely bounded cohomolog…
We prove a basic result about tensor products of a $\text{II}_1$ factor with a finite von Neumann algebra and use it to answer, affirmatively, a question asked by S. Popa about maximal injective factors.
Classification of Invariant Subalgebras in a class of factors with property (T)
本文研究了具有性质(T)的群$G_n$在因子$L(G_n)$中的不变子代数分类，针对$n=2$和$n\geq 3$分别给出了不同结论，并推导出$L(G_n)$具有唯一最大Haagerup不变子代数的结论。
In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse free product components. In the type III setting, this is the first classification result for infinite tensor product non-amenable factors. Our proof is based on Popa's intertwining techniques and a characterization of relative amenability on the continuous cores.
Gromov constructed uncountably many pairwise non-isomorphic discrete groups with Kazhdan's property (T). We will show that no separable II_1-factor can contain all these groups in its unitary group. In particular, no separable II_1-factor can contain all separable II_1-factors in it. We also show that the full group C*-algebras of some of these groups fail the lifting property.
We show that the discrete duals of the universal unitary quantum groups and orthogonal quantum groups have Kirchberg's factorization property when n is different from 3.
A locally compact group $G$ has the factorization property if the map $$C^*(G)\odot C^*(G)\ni a\otimes b\mapsto λ(a)ρ(b)\in\mathcal B(L^2(G))$$ is continuous with respect to the minimal C*-norm. This paper seeks to initiate a rigorous study of this property in the case of locally compact groups which, in contrast to the discrete case, has been relatively untouched. A partial solution to the question of when the factorization property passes to continuous embeddings is given -- a question which traces back to Kirchberg's seminal work on the topic and is known to be false in general. It is also …
In this article we provide the first examples of property (T) $\rm II_1$ factors $\mathcal N$ with trivial fundamental group, $\mathcal F (\mathcal N)=1$. Our examples arise as group factors $\mathcal N=\mathcal L(G)$ where $G$ belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in \cite{Va04} and the ones introduced recently in \cite{CDK19} using the Belegradek-Osin Rips construction from \cite{BO06}. In particular, our results provide a continuum of explicit pairwise non-isomorphic property (T) factors.
We show that the amenability of a locally compact group $G$ is equivalent to a factorization property of $VN(G)$ which is given by $ VN(G) = <VN(G)^*VN(G)>$. This answer partially two problems proposed by Z. Hu and M. Neufang in their article \textit{Distinguishing properties of Arens irregularity}, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1753-1761.
Techniques introduced by G. Pisier in his proof that finite von Neumann factors with property gamma have length at most 5 are modified to prove that the length is 3. It is proved that if such a factor is a complemented subspace of some larger C*-algebra then there exists a projection of norm one from the larger onto the smaller algebra. A new proof of the fact that the second continuous Hochschild cohomology group of such an algebra with coefficients in the algebra vanishes, is also included.
In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question of Elek and Szabo. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in an assertion of Kirchberg and provide an example of a group which does not have the factorization property and is still a subgro…
We introduce the notion of a generalized Jung factor: a II$_1$ factor $M$ for which any two embeddings of $M$ into its ultrapower $M^{\mathcal U}$ are equivalent by an automorphism of $M^{\mathcal U}$. We show that $\mathcal R$ is not the unique generalized Jung factor but is the unique $\mathcal R^{\mathcal U}$-embeddable generalized Jung factor. We use model-theoretic techniques to obtain these results. Integral to the techniques used is the result that if $M$ is elementarily equivalent to $\mathcal R$, then any elementary embedding of $M$ into $\mathcal R^{\mathcal U}$ has factorial relativ…
We prove that every separable tracial von Neumann algebra embeds into a II$_1$ factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. These results are obtained by using the class of wreath-like product groups introduced recently in \cite{CIOS21}.
An exotic II$_1$ factor without property Gamma
本文通过新的迭代融合自由积构造出第一个不具有 Gamma 性质且不与自由群因子或矩阵代数超积初等等价的 II$_1$ 因子，并给出两个无 Gamma 性质但互不初等等价的显式例子。
We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff II_1 factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non implications were obtained in [Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining implications and counterexamples.
Property (T) group factors whose Jones index set equals all positive integers
本文通过结合变形/刚性理论、子因子理论与几何群论，构造出一类性质（T）广义花环式积群的群冯·诺依曼代数，证明其琼斯指标集恰为全体正整数，并由此得到连续统多个两两非稳定同构的因子，推进了 de la Harpe 的公开问题。
We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg's theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness. We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (facto…
Hypertrace and entropy gap characterizations of property (T) for $\mathrm{II}_1$ factors
本文通过构造包含 $\mathrm{II}_1$ 因子 $M$ 的 von Neumann 代数，利用超迹、Furstenberg 熵和熵间隙给出性质 (T) 与 Haagerup 性质的等价刻画。
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $…
We show that any infinite collection $(Γ_n)_{n\in \mathbb N}$ of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic \emph{infinite product rigidity} phenomenon. If $Λ$ is an arbitrary group such that $L(\oplus_{n\in \mathbb N} Γ_n)\cong L(Λ)$ then there exists an infinite direct sum decomposition $Λ=(\oplus_{n \in \mathbb N} Λ_n )\oplus A$ with $A$ icc amenable such that, for all $n\in \mathbb N$, up to amplifications, we have $L(Γ_n) \cong L(Λ_n)$ and $L(\oplus_{k\geq n} Γ_k )\cong L((\oplus_{k\geq n} Λ_k) \oplus A)$. The result is sharp and complements the pre…
A character approach to the ISR property
本研究提出研究ISR性质的特征标方法，引入不可分解正则特征标性质推导相关群的ISR性质，证明近似有限群具有ISR性质，并构造出两类新的具有ISR性质的非顺从群。
On ultraproduct approximations and property (T) factors
本文提出框架，利用变形/刚性理论关键方面研究II₁因子的连续模型理论，解决多个开放问题，如证明某些群 von Neumann 代数非元素等价，以及非伪矩阵代数，还提供无限非元素等价满因子族。
In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type ${\rm II}\sb 1$ factors and $M_n(\cc)$) and symmetric gauge norms on $L^\infty[0,1]$ and $\cc^n$. As the first application, we obtain that the class of unitarily invariant norms on a type ${\rm II}\sb 1$ factor coincides with the class of symmetric gauge norms on $L^\infty[0…
One of the earliest invariants introduced in the study of finite von Neumann algebras is the property Gamma of Murray and von Neumann. In this note we prove that it is not possible to classify separable $\rm{II}_1$ factors satisfying the property Gamma up to isomorphism by a Borel measurable assignment of countable structures as invariants. We also show that the same holds true for the full $\rm{II}_1$ factors.
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.
We prove that any II$_1$ factor arising from a free ergodic probability measure preserving action $Γ\curvearrowright X$ of a product $Γ=Γ_1\times\dots\timesΓ_n$ of icc hyperbolic, free product or wreath product groups is prime, provided $Γ_i\curvearrowright X$ is ergodic, for any $1\leq i\leq n.$ We also completely classify all the tensor product decompositions of a II$_1$ factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II$_1$ factors. Finally…
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