It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x)\to 0$ as $t\to\infty$ for every $x\in E$, where $μ_t(x)$ is the non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric spaces for which Dunford-Schwartz-type individual ergodic theorems hold is outlined.
We studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier's interpolation theorem for this case.
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.
In this paper, we provide a generalized version of the Voiculescu theorem for normal operators by showing that, in a von Neumann algebra with separable pre-dual and a faithful normal semifinite tracial weight $τ$, a normal operator is an arbitrarily small $(\max\{\|\cdot\|, \Vert\cdot\Vert_{2}\})$-norm perturbation of a diagonal operator. Furthermore, in a countably decomposable, properly infinite von Neumann algebra with a faithful normal semifinite tracial weight, we prove that each self-adjoint operator can be diagonalized modulo norm ideals satisfying a natural condition.
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric …
We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and Rényi's relative entropy known in finite dimension.
We extend Gour et al's characterization of quantum majorization via conditional min-entropy to the context of semifinite von Neumann algebras. Our method relies on a connection between conditional min-entropy and operator space projective tensor norm for injective von Neumann algebras. This approach also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to noncommutative vector-valued $L_1$-space.
We introduce a class of unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norms with repect to $τ$ on a von Neumann algebra $\mathcal{M}$ with a faithful, normal, semifinite tracial weight $τ$. We prove a Beurling-Chen-Hadwin-Shen theorem for $H^\infty$-invariant spaces of $L^α(\mathcal{M},τ)$, where $α$ is a unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norm with respect to $τ$, and $H^\infty$ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the $H^\infty$-invariant subspaces of a noncommutati…
Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},τ)$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $δ:\mathcal{A}\to E(\mathcal{M},τ)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.
In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let $\mathcal{M}$ be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $τ$ be a faithful normal semifinite tracial weight of $\mathcal M$. Suppose that $H$ and $H_1$ are self-adjoint operators affiliated with $\mathcal{M}$. We show that if $H-H_1$ is in $\mathcal{M}\cap L^{1}\left(\mathcal{M},τ\right)$, then the ${norm}$ absolutely continuous parts of $H$ and $H_1$ are unitarily equivalent. This implies that the real par…
In this paper we extend the Pietsch correspondence for ideals of compact operators and traces on them to the semifinite setting. We prove that a shift-monotone space $E(\Z)$ of sequences indexed by $\Z$ defines a Calkin space $E(\cM,τ)$ of $τ$-measurable operators affiliated with a semifinite von Neumann algebra $\cM$ equipped with a faithful normal semifinite trace $τ$. Furthermore, we show that shift-invariant functionals on $E(\Z)$ generate symmetric functionals on $E(\cM,τ)$. In the special case, when the algebra $\cM$ is atomless or atomic with atoms of equal trace, the converse also hold…
In 2008, Blecher and Labuschagne extended Beurling's classical theorem to $H^\infty$-invariant subspaces of $L^p(\mathcal{M},τ)$ for a finite von Neumann algebra $\mathcal{M}$ with a finite, faithful, normal tracial state $τ$ when $1\le p\le \infty$. In this paper, using Arveson's non-commutative Hardy space $H^\infty$ in relation to a von Neumann algebra $\mathcal{M}$ with a semifinite, faithful, normal tracial weight $τ$, we prove a Beurling-Blecher-Labuschagne theorem for $H^\infty$-invariant spaces of $L^p(\mathcal{M},τ)$ when $0<p\leq\infty$. The proof of the main result relies on proofs …
Group amenability properties for von Neumann algebras
在他对顺从的酉表示的研究中，M. E. B.贝卡问是否有一个类似的代表性显着的定点财产顺从的群体。在本文中，我们证明了这样一个不动点定理在更一般的背景下的$G$ -顺从冯诺依曼代数$M$，其中$G$是一个局部紧群作用在$M$。本文将Connes和Bekka的Følner条件推广到$M$是半有限的，并且有一个忠实的、半有限的、正规的迹在$G$作用下不变的情形.
Banach non-isomorphism of type III and semifinite noncommutative $L_p$-spaces
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学学院）；School of Mathematics and Statistics, University of NSW（新南威尔士大学数学与统计学院）
In [10], Halmos proved an interesting result that the set of irreducible operators is dense in $\mathcal B(\mathcal H)$ in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra $\mathcal M$ with separable predual, an operator $a\in \mathcal M$ is said to be {irreducible in} $\mathcal M$ if $W^*(a)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(a)'\cap \mathcal M={\mathbb C} \cdot I$. In this paper, let $Φ(\cdot)$ be a $\Vert\cdot\Vert$-dominating, unitarily invariant norm (see Definition 2.1), where by $\Vert\cdot\Vert$ we denote the operator norm. We prove that in eve…
We show that ergodic flows in noncommutative fully symmetric spaces (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also discussed.
In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the…
Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $τ$. Let $E(\mathcal{M},τ) $ be a symmetric operator space affiliated with $ \mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},τ)$. We obtain general description of all bounded hermitian operators on $E(\mathcal{M},τ)$. This is the first time that the description of he…
We prove that the predual of any von Neumann algebra is $1$-Plichko, i.e., it has a countably $1$-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U.~Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the self-adjoint part of the predual is $1$-Plichko as well.
On the structure of isometries between noncommutative Lp spaces
证明了与von Neumann代数相关的非交换Lp空间之间的等距的一些结构结果。我们发现，当M_1具有Watanabe扩张性质的变体时，等距T：Lp（M_1）to Lp（M_2）（1 le p < infty，p not 2）可以正则地表示为某种简单形式.相反，如果M_1是“近似半有限的”（定义如下），这种形式总是定义等距。虽然这两个性质都没有被完全理解，我们表明，他们享有的所有半有限代数和超有限代数（没有I_2型的和项），加上其他。因此，该分类比Yeadon的半有限代数定理（及其最近在[JRS]中的改进）更强，并且证明使用了独立的技术。与此相关的是，我们研究的模块化理论的正投影从一个冯诺依曼代数到约旦图像的另一个冯诺依曼代数，并使用这样的投影构造新的Lp等距插值。一些补充的结果和问题也被提出。
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
This article is devoted to studying individual ergodic theorems for subsequential weighted ergodic averages on the noncommutative Lp-spaces associated to a semifinite von Neumann algebra M. In particular, we establish the convergence of these averages along sequences with density one and certain types of block sequences with positive lower density, and we extend known results along uniform sequences in the sense of Brunel and Keane.
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\M$. For $0<p \leq\infty$, let $\h_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$ and the index $p$. We prove that for $0<p<\infty$, the compatible couple $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ is $K$-closed in the couple $\big(L_p(\mathcal{N}), L_\infty(\mathcal{N}) \big)$ for an appropriate amplified semifinite von …
We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.
Jensen's inequality for partial traces in von Neumann algebras
Wallenberg Centre for Quantum Technology, Chalmers University of Technology（瓦伦堡量子技术中心，查尔姆斯理工大学）；Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学数学科学系）
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal weight $ϕ$ and $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$ such that the restriction of $ϕ$ to $\mathcal{N}$ is semifinite and such that $\mathcal{N}$ is invariant by the modular group of $ϕ$. Let $\mathcal{E}$ be the weight preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality: \[\|x\|_p^2\ge\bigl \|\mathcal{E}(x)\bigr\|_p^2+(p-1)\bigl\|x-\mathcal{E}(x)\bigr\|_p^2, \qquad x\in L_p(\mathcal{M}),1<p\le2,\] which extends the celebrated Ball-Ca…
