Liming Ge
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.
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Let M_i be a von Neumann algebra, and B_i be a maximal injective von Neumann subalgebra of M_i, i=1,2. If M_1 has separable predual and the center of B_1 is atomic, e.g., B_1 is a factor, then B_1\tensor B_2 is a maximal injective von Neuamnn subalgebra of M_1\tensor M_2. This partly answers a question of Popa
In this article, we proved the following results. Let $L(F(n_i))$ be the free group factor on $n_i$ generators and $λ(g_{i})$ be one of standard generators of $L(F(n_i))$ for $1\le i\le N$. Let $\A_i$ be the abelian von Neumann subalgebra of $L(F(n_i))$ generated by $λ(g_{i})$. Then the abelian von Neumann subalgebra $\otimes_{i=1}^N\A_i$ is a maximal injective von Neumann subalgebra of $\otimes_{i=1}^N L(F(n_i))$. When $N$ is equal to infinity, we obtained McDuff factors that contain maximal injective abelian von Neumann subalgebras.
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…
Let $\M$ be a von Neumann algebra acting on a Hilbert space $\H$, and $\N$ be a singular von Neumann subalgebra of $\M.$ If $\N\tensor\B(\K)$ is singular in $\M\tensor\B(\K)$ for any Hilbert space $\K$, we say $\N$ is \emph{completely singular} in $\M$. We prove that if $\N$ is a singular abelian von Neumann subalgebra or if $\N$ is a singular subfactor of a type $II_1$ factor $\M$, then $\N$ is completely singular in $\M$. For any type $II_1$ factor $\M$, we construct a singular von Neumann subalgebra $\N$ of $\M$ ($\N\neq \M$) such that $\N\tensor\B(l^2(\mathbb{N}))$ is regular (hence not si…
In this paper, we exhibit strongly singular maximal abelian subalgebras living inside certain k-folded tensors of von Neumann group factors. The two classes of groups under consideration are the free groups of rank greater than 2 and the free product of m>2 groups, all finite and having the same order p>2 (m>p-1). We prove actually more, namely that the unique trace preserving conditional expectations from these group factors onto their Laplacian subalgebras are asymptotic homomorphisms, which in turn implies that the subalgebras are strongly singular masas.
Let $M$, $N$, $R$ be $W^{*}$--algebras, with $R$ unitally embedded in both $M$ and $N$. by using Reduction Theory, we extend the previous description of the $W^{*}$--tensor product $M\bar\otimes_{R}N$ over the common $W^{*}$--subalgebra $R$ and its predual $(M\bar\otimes_{R}N)_{*}$ to the $σ$--finite case.
In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the k-folded tensor product of the free group factor (on N>1 generators). A. Sinclair and R. Smith introduced the class of strongly singular maximal abelian von Neumann algebras (MASA) of type II_1 factors. One of their examples was the Laplacian subalgebra of the free group factor, known to be a singularMASA. Using the techniques presented by A. Sinclair and R.Smith, we show that the Laplacian subalgebra of the k-folded tensor product of the free group factor with itself is a strongly singular…
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspo…
Invariant von Neumann subalgebras of crossed products and tensor products
该文研究可数离散群保迹作用下交叉积与张量积的不变冯·诺依曼子代数，在特定假设下分类其结构，并构造了自由群的相关保测作用实例。
In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l- and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and Müller, and proves a conjecture of the second named author. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor pro…
Relations amongst the distances between $C^{*}$-subalgebras and some canonically associated operator algebras
本文证明两个$C^*$-子代数的Christensen距离和Kadison-Kastler距离分别等于其包络von Neumann代数及与任意单位交换$C^*$-代数的最小张量积代数之间的距离。
We investigate spectral properties of the tensor products of two quantum channels defined on matrix algebras. This leads to the important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ the multiplicative domain of $\mathcal{E}_1\otimes\mathcal{E}_2$ splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of ma…
We define centrally large subalgebras of simple unital C*-algebras, strengthening the definition of large subalgebras in previous work. We prove that if A is any infinite dimensional simple separable unital C*-algebra which contains a centrally large subalgebra with stable rank one, then A has stable rank one. We also prove that large subalgebras of crossed product type are automatically centrally large. We use these results to prove that if X is a compact metric space which has a surjective continuous map to the Cantor set, and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable r…
On stability of distance under some tensor products and some calculations
本文研究了某些张量积下的距离稳定性，并对交叉积算子代数的子代数间距离进行了具体计算。
We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,μ)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.
Frobenius subalgebra lattices in tensor categories
本文将Watatani的中间子因子有限性定理推广到更广泛的单子范畴中，通过引入环境自对偶的概念，刻画了阿贝尔单子范畴中弗罗贝尼乌斯子代数偏序格子的子格，并在弱正性框架下证明了任何连通弗罗贝尼乌斯代数的相干子格都是有限的，从而显著扩展了Watatani定理。
We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a…
An algebra extension $A \| B$ is right depth two in this paper if its tensor-square is $A$-$B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $A$ with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory math.RA/0108067 but extends the main theorem of depth two theory, as for example in math.RA/0107064. That is, a right depth two extension has right bialgebroid T = (A …
Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
该论文将条件熵对偶性推广至任意冯·诺依曼代数子代数，并建立无限维量子斯坦引理，为子代数相对熵提供复合假设检验中的操作解释。
Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group $Γ$ its von Neumann algebra $L(Γ)$ satisfies the so-called ISR property: \emph{any von Neumann subalgebra $N\subseteq L(Γ)$ that is normalized by all group elements in $Γ$ is of the form $N= L(Σ)$ for a normal subgroup $Σ\lhd Γ$.} In particular, this applies to all groups $Γ$ in each of the following classes: all icc (relatively) hyperbolic groups, most mapping class groups of surfaces, all outer automorphisms of free groups with at le…
We define and systematically study nonassociative C*-algebras as C*-algebras internal to a topological tensor category. We also offer a concrete approach to these C*-algebras, as G-invariant, norm closed *-subalgebras of bounded operators on a G-Hilbert space, with deformed composition product. Our central results are those of stabilization and Takai duality for (twisted) crossed products in this context.
Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version
该研究针对离散群作用下的$C^*$-代数，给出中间子代数为张量积的抽象条件，推广经典分裂结果，证明拓扑版中间因子定理，引入一致刚性作用概念并给出多类满足条件的不变中间代数包含关系。
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