Weichen Gao, Marius Junge, Weichen Gao
In this paper we study conditions under which, for an inclusion of finite von Neumann algebras $N \subseteq M$, we have the reduced amalgamated free product $\ast_N M$ is embeddable into $(R \bar{\otimes} N_1)^ω$ for some other finite von Neumann algebra $N_1$, where $R$ is the hyperfinite $\textrm{II}_1$ factor.
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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.
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.
Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
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 investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebras $M = M_1 \ast_B M_2$. Our main result states that under natural analytic assumptions, any amenable subalgebra of $M$ that has a large intersection with $M_1$ is actually contained in $M_1$. The proof does not rely on Popa's asymptotic orthogonality property but on the study of non normal conditional expectations.
We investigate the use of Popa's asymptotic orthogonality to establish maximal amenability for amalgamated free product von Neumann algebras. Although new techniques have recently been developed to consider amalgamated free products, we find that the technique developed by Popa in the 1980s can be used to demonstrate maximal amenability for a certain family of amalgamated free product von Neumann algebras.
We provide a fairly large family of amalgamated free product groups $Γ=Γ_1\ast_ΣΓ_2$ whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that $Γ_i$ is a product of two icc non-amenable bi-exact (e.g., hyperbolic) groups, and $Σ$ is icc amenable and has trivial one-sided commensurator in $Γ_i$, for every $i\in\{1,2\}$. Then $Γ$ satisfies the following rigidity property: any group $\La$ such that $L(\La)$ is isomorphic to $L(\G)$ admits an amalgamated free product decomposition $\La=\La_1\ast_Δ\La_2$ such that the inclusions $L(Δ)\subseteq L…
For any reduced amalgamated free product $\mathrm{C}^*$-algebra $(A,E)=(A_1, E_1) \ast_D (A_2,E_2)$, we introduce and study a canonical ambient $\mathrm{C}^*$-algebra $Δ\mathbf{T}(A,E)$ of $A$ which generalizes the crossed product arising from the canonical action of an amalgamated free product group on the compactification of the associated Bass--Serre tree. Using an explicit identification of $Δ\mathbf{T}(A,E)$ with a Cuntz--Pimsner algebra we prove two kinds of "amenability" results for $Δ\mathbf{T}(A,E)$; nuclearity and universality. As applications of our framework, we provide new concept…
We investigate factoriality, Connes' type ${\rm III}$ invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type ${\rm III}$ invariants.
On Relative Biexactness of Amalgamated Free Product von Neumann Algebras
本文证明具有公共内射融合子代数的弱正合迹冯·诺依曼代数的融合自由积在相应条件下具有相对双正合性，并由此推广了群情形下的结构分解与子代数吸收定理。
We show that a free product of a II_1-factor and a finite von Neumann algebra with amalgamation over a finite dimensional subalgebra is always a II_1-factor, and provide an algorithm for describing it in terms of free products (with amalgamation over the scalars) and compression/dilation. As an application, we show that the class of direct sums of finitely many von Neumann algebras that are interpolated free group factors, hyperfinite II_1-factors, type I_n algebras for n finite, and finite dimensional algebras, is closed under taking free products with amalgamation over finite dimensional sub…
With a minor change made in the previous construction we observe that any reduced HNN extension is precisely a compressed algebra of a certain reduced amalgamated free product in both the von Neumann algebra and the $C^*$-algebra settings. It is also pointed out that the same fact holds true even for universal HNN extensions of C$^*$-algebras. We apply the observation to the questions of factoriality and type classification of HNN extensions of von Neumann algebras and also those of simplicity and $K$-theory of (reduced or universal) HNN extensions of $C^*$-algebras.
Let $(M_i)_{i}$ be a (finite or infinite) family of finite von Neumann algebras with a common subalgebra $P$. When $φ:\IN\rightarrow\IC$ is a function, we define the radial multiplier $M_φ$ on the amalgamated free product $M=M_1\free_P M_2\free_P\ldots$ setting $M_φ(x)=φ(n)x$ for every reduced expression $x$ of length $n$. In this paper we give a sufficient condition on $φ$ to ensure that the corresponding radial multiplier $M_φ$ is a completely bounded map, and moreover we give an upper bound on its completely bounded norm. Our condition on $φ$ does not depend on the choice of von Neumann alg…
Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group Γ_1 * Γ_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer N_M(A)'' of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M_1 *_B M_2. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in crossed products by Γ_1 * Γ_2.
Connes-embeddability and twisted group von Neumann algebras
本文定义余循环-超线性群，证明其包含特定扩张与中心扩张，并在多种群运算下封闭，从而推广 Connes-可嵌入性。
We establish a platform to transfer $L_p$-completely bounded maps on tensor products of von Neumann algebras to $L_p$-completely bounded maps on the corresponding amalgamated free products. As a consequence, we obtain a Hörmander-Mikhlin multiplier theory for free products of groups. Let $\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1, g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$ satisfying the classical Hörmander-Mikhlin condition, the linear map $M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by $λ(g)\mapsto m(k_1,\cdo…
We calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von Neumann algebras, with amalgamation over a hyperfinite subalgebra. In particular, some `exotic' Popa algebra generators of free group factors are shown to have the expected free entropy dimension. We also show that microstates and non--microstates free entropy dimension agree for generating sets of many groups. In the appendix by Wolfgang Lueck, the first L^2-Betti number for certain amalgamated free products of groups is calculated.
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
Let $\mathcal{M}_i$ be a family of $\mathrm{II}_1$-factors, containing a common $\mathrm{II}_1$-subfactor $\mathcal{N}$, such that $[\mathcal{M}_i:\mathcal{N}] \in \mathbb{N}_0$ for all $i$. Furthermore, let $ϕ\colon \mathbb{N}_0 \to \mathbb{C}$. We show that if a Hankel matrix related to $ϕ$ is trace-class, then there exists a unique completely bounded map $M_ϕ$ on the amalgamated free product of the $\mathcal{M}_i$ with amalgamation over $\mathcal{N}$, which acts as an radial multiplier. Hereby we extend a result of U. Haagerup and the author for radial multipliers on reduced free products o…
We establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF.
We obtain new Bass-Serre type rigidity results for ${\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras $\mathcal{M}_1 \ast_B \mathcal{M}_2$ over an abelian von Neumann algebra $B$, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\rm II_1}$ and in the type ${\rm III}$ case, new examples of prime factors.
Given reduced amalgamated free products of C$^*$-algebras, $(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_i\to D_i$. This result is extended to show the existance of the reduced amalgamated free product of certain classes of unital completely positive maps. Analogues of the above mentioned results are proved for von Neumann algebras.
Using a non-negative curvature condition, we prove the complete version of modified log-Sobolev inequalities for central Markov semigroups on various compact quantum groups, including group von Neumann algebras, free orthogonal group and quantum automorphism groups. We also prove that the "geometric Ricci curvature lower bound" introduced by Junge-Li-LaRacuente is stable under tensor products and amalgamated free products. As an application, we obtain the geometric Ricci curvature lower bound and complete modified logarithmic Sobolev inequality for word-length semigroups on free group factors …
We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property fo…
Irreducibility and weak spectral gap in free product von Neumann algebras
该研究针对非平凡冯·诺依曼代数的自由积，利用融合自由积技术建立其自由积子代数及中心化子子代数的不可约性与弱谱隙结果，还得到了冯·诺依曼代数图积的类似不可约性结果。
Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical th…
We consider crossed product von Neumann algebras arising from free Bogoljubov actions of the integers. We describe several presentations of them as amalgamated free products and cocycle crossed products and give a criterion for factoriality. A number of isomorphism results for free Bogoljubov crossed products are proved, focusing on those arising from almost periodic representations. We complement our isomorphism results by rigidity results yielding non-isomorphic free Bogoljubov crossed products and by a partial characterisation of strong solidity of a free Bogoljubov crossed products in term…
On some free products of von Neumann algebras which are free Araki-Woods factors
本文证明了I型因子等von Neumann代数在非迹几乎周期态下的自由积是自由Araki-Woods因子，其完全分类由Connes的Sd不变量给出，并回答了Dykema和Shlyakhtenko的相关问题。
We present families of pairs of finite von Neumann algebras $A\subset M$ where $A$ is a maximal injective masa in the type $\mathrm{II}_1$ factor $M$ with separable predual. Our results make use of the strong mixing and the asymptotic orthogonality properties of $A$ in $M$ and are borrowed from ideas of S. Popa who proved that if $G$ is a non abelian free group and if $a$ is one of its generators, then the von Neumann algebra generated by $a$ is maximal injective in the factor $L(G)$. Our results apply to pairs $H<G$ where $H$ is an infinite abelian subgroup of a suitable amalgamated product g…
We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This implies that the decomposition of the free group $\F_\infty$ into reduced words starting with distinct free generators is completely unconditional in $L^p$. We study the case of Voiculescu's amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness-problem posed by Ozawa, a length independent estimat…
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $Γ$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a free ergodic probability measure preserving action of $Γ$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability …
Let $f_1, f_2, ..., f_n$ be a family of independent copies of a given random variable f in a probability space $(Ω, \mathcal{F}, μ)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$ $$\Big(\int_Ω \Big[ \sum_{k=1}^n |f_k|^q \Big]^{\frac{p}{q}} d μ\Big)^{\frac1p} \sim \max_{r \in \{p,q\}} {n^{\frac1r} \Big(\int_Ω|f|^r dμ\Big)^{\frac1r}}.$$ We prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probab…
In the mid thirties Murray and von Neumann found a natural way to associate a von Neumann algebra $L(Γ)$ to any countable discrete group $Γ$. Classifying $L(Γ)$ in term of $Γ$ is a notoriously complex problem as in general the initial data tends to be lost in the von Neumann algebraic regime. An important problem in the theory of von Neumann algebras is to completely describe all possible tensor decompositions of a given group von Neumann algebra $L(Γ)$. In this direction the main goal is to investigate how exactly a tensor decomposition of $L(Γ)$ relates to the underlying group $Γ$. In this d…
Exotic full factors via weakly coarse bimodules
通过弱粗双模构造完整III_1因子，无需依赖Property (T)现象。
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。
Coarse rigidity in von Neumann algebras via derivations
本文通过模导子提出冯·诺依曼代数的相对粗刚性概念，扩展L²-刚性至非迹情形，并证明无相对amenable项时形变收敛，从而统一证明完全性等不可分解性质。
We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$ and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into one of the $M_i$'s. We apply this to the case $M_i$ are w-rigid II$_1$ factors, with $B$ equal to either $\Bbb C$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, à la Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other similar …
We canonically associate to any planar algebra two type II_{\infty} factors M_{+} and M_{-}. The subfactors constructed previously by the authors in a previous paper are isomorphic to compressions of M_{+} and M_{-} to finite projections. We show that each \mathfrak{M}_{\pm} is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M_{+} \cong M_{-} is the amplification a free group factor on a finite number of generators. As an applic…
Relative biexactness and mixing in von Neumann algebras
提出一种新技术，将相对双精确性提升为一般von Neumann代数的双精确性，应用于融合自由积和图积，推广了Caspers-Borst和Blufstein-Goldman-Oyakawa的结果。
Asymptotic freeness in tracial ultraproducts
本文证明了迹超乘积冯·诺伊曼代数中关于相对换位子的渐近自由性结果，并基于此推导出一般吸收结论与新的提升定理，构造了首个不具有性质Gamma且不与任何弥散迹冯·诺伊曼代数自由乘积初等等价的II_1因子实例。
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…
We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to $\varepsilon$-free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular $\varepsilon$ arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary $\varepsilon$ may be realized in this way. As a result we obtain a new proof that $\mathcal{R}^ω$-embeddability is preserved under graph products of von Neumann algebras, along with an explicit recip…
Generalized Poincaré inequality for quantum Markov semigroups
本文提出了一种针对量子马尔可夫半群的广义皮卡德不等式，通过非交换导数和链规则估计，推广了半交换结果，并应用于非交换辛钦和亚指数集中不等式的恢复。
Let $\mathcal{A}$ denote the reduced amalgamated free product of a family $\mathsf{A}_1, \mathsf{A}_2, ..., \mathsf{A}_n$ of von Neumann algebras over a von Neumann subalgebra $\Be$ with respect to normal faithful conditional expectations $\Es_k: \mathsf{A}_k \to \Be$. We investigate the norm in $L_p(\Al)$ of homogeneous polynomials of a given degree $d$. We first generalize Voiculescu's inequality to arbitrary degree $d \ge 1$ and indices $1 \le p \le \infty$. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from wh…
We introduce symmetric states and quantum symmetric states on universal unital free product C*-algebras an arbitrary unital C*-algebra A with itself infinitely many times, as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove existence of conditional expectations onto tail algebras in various settings and we define a natural C*-subalgebra of the tail algebra, called the tail C*-algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Koestler and Speicher, we prove a de Finetti type theorem that characterizes quan…
Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects. These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$, depending on whether the spectrum of the category is finite or infinite) or they can be of type III$_λ$, $λ\in (0,1]$. The choice of type is tuned by the choice of Tomita structure (defined in the paper) on certain bimodules we use in the construction. Moreover, if the spectrum is infinite we realize the whole tensor category directly as endomorphisms of th…
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