Operator Algebras

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Showing new listings for Tuesday, 30 September 2025

New submissions (showing 1 of 1 entries )

[1] arXiv:2509.24145 [cn-pdf, pdf, html, other]

Title: Spectral continuity for étale groupoids with the Rapid decay property Show Chinese title

Title: 谱连续性对于具有快速衰减性质的层叠群胚

Tom Stoiber

Subjects: Operator Algebras (math.OA)

We show that the reduced groupoid C*-algebras of continuous fields of \'etale groupoids satisfying the rapid decay property yield continuous fields of C*-algebras. This establishes a new sufficient criterion that applies in the non-amenable case where the full and reduced groupoid algebras may differ. Potential applications include convergence of spectra in inverse systems of finite-index subgroups and magnetic models on hyperbolic lattices. Show Chinese abstract

我们证明了满足快速衰减性质的连续条痕层的约化群胚C*-代数，产生连续的C*-代数层。 这建立了一个新的充分条件，在非可约情况下适用，此时全群胚代数和约化群胚代数可能不同。 潜在的应用包括有限指数子群逆系统的谱收敛以及双曲格点上的磁模型。

Cross submissions (showing 6 of 6 entries )

[2] arXiv:2509.22743 (cross-list from math.FA) [cn-pdf, pdf, html, other]

Title: Multiplicative trace and spectrum preservers on stochastic matrices Show Chinese title

Title: 乘法迹和谱保全映射在随机矩阵上

Ming-Cheng Tsai, Huajun Huang

Comments: 31 pages

Subjects: Functional Analysis (math.FA) ; Operator Algebras (math.OA) ; Probability (math.PR)

We characterize maps $\phi_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property: \begin{eqnarray*} \textrm{spec} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad \textrm{tr} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m), \end{eqnarray*} where $\mathcal{S}$ is the set of $n\times n$ doubly stochastic, row stochastic, or column stochastic matrices, or the space spanned by one of these sets. Linearity is assumed when $m=1$. We show that every stochastic matrix contains a real doubly stochastic component that carries the spectral information. In consequence, the multiplicative spectrum or trace preservers on these sets $ \mathcal{S} $ are linked to the corresponding preservers on the space of doubly stochastic matrices. Moreover, when $m\ge 3$, multiplicative trace preservers always coincide with multiplicative spectrum preservers. Show Chinese abstract

我们表征具有乘法谱或迹保持性质的映射$\phi_i: \mathcal{S} \to \mathcal{S}$、$i=1, \ldots, m$和$m\ge 1$： \begin{eqnarray*} \textrm{spec} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad \textrm{tr} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m), \end{eqnarray*} 其中$\mathcal{S}$是$n\times n$双随机、行随机或列随机矩阵的集合，或者这些集合之一所张成的空间。 当$m=1$时假设线性性。 我们证明每个随机矩阵都包含一个实双随机分量，该分量携带谱信息。 因此，这些集合上的乘法谱或迹保持映射$ \mathcal{S} $与双随机矩阵空间上的相应保持映射相关。此外，当$m\ge 3$时，乘法迹保持映射总是与乘法谱保持映射一致。

[3] arXiv:2509.23380 (cross-list from math.KT) [cn-pdf, pdf, html, other]

Title: Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity Show Chinese title

Title: 商Roe代数上的相对高指标理论与无穷远处的正标量曲率

Liang Guo, Qin Wang, Chen Zhang

Subjects: K-Theory and Homology (math.KT) ; Functional Analysis (math.FA) ; Operator Algebras (math.OA)

In this paper, we employ quotients of Roe algebras as index containers for elliptic differential operators to study the existence problem of Riemannian metrics with positive scalar curvature on non-compact complete Riemannian manifolds. The non-vanishing of such an index locates the precise direction at infinity of the obstructions to positive scalar curvature, and may be viewed as a refinement of the positive scalar curvature problem. To achieve this, we formulate the relative coarse Baum-Connes conjecture and the relative coarse Novikov conjecture, together with their maximal versions, for general metric spaces as a program to compute the $K$-theory of the quotients of the Roe algebras relative to specific ideals. We show that if the metric space admits a relative fibred coarse embedding into Hilbert space or an $\ell^p$-space, certain cases of these conjectures can be verified, which yield obstructions to the existence of uniformly positive scalar curvature metrics in specified directions at infinity. As an application, we prove that the maximal coarse Baum-Connes conjecture holds for finite products of certain expander graphs that fail to admit fibred coarse embeddings into Hilbert space. Show Chinese abstract

在本文中，我们采用Roe代数的商作为椭圆微分算子的指标容器，以研究非紧致完备黎曼流形上具有正标量曲率的黎曼度量的存在性问题。 这种指标的非零性定位了正标量曲率障碍在无穷远处的精确方向，并可能被视为正标量曲率问题的一种改进。 为了实现这一点，我们为一般的度量空间提出了相对粗略的Baum-Connes猜想和相对粗略的Novikov猜想，以及它们的最大版本，作为计算Roe代数关于特定理想的商的$K$-理论的计划。 我们证明，如果度量空间可以相对纤维化粗略嵌入到希尔伯特空间或$\ell^p$-空间，则这些猜想的某些情况可以被验证，这会导致在无穷远处指定方向上存在均匀正标量曲率度量的障碍。 作为应用，我们证明了最大粗略Baum-Connes猜想对于某些失败嵌入到希尔伯特空间的展开图的有限乘积成立。

[4] arXiv:2509.23539 (cross-list from math.FA) [cn-pdf, pdf, html, other]

Title: Noncommutative localizations and joint spectra of a contractive quantum plane Show Chinese title

Title: 非交换局部化和压缩量子平面的联合谱

Anar Dosi

Subjects: Functional Analysis (math.FA) ; Operator Algebras (math.OA)

In the present paper we investigate the localizations in the sense of J. L. Taylor of the Arens-Michael-Fr\'echet algebras occurred within the formal geometry of a contractive q-plane. It turns out that all noncommutative Fr\'echet algebras obtained by the Fr\'echet sheaves of the formal geometry are indeed localizations. That topological homology property of the structure sheaf results in the key properties of Taylor and Putinar spectra of the Banach q-modules. Show Chinese abstract

在本文中，我们研究了在J. L. Taylor意义下的局部化，这些局部化出现在收缩q平面的形式几何中的Arens-Michael-Fréchet代数。结果表明，通过形式几何的Fréchet层得到的所有非交换Fréchet代数确实是局部化。结构层的这一拓扑同调性质导致了Banach q模的Taylor和Putinar谱的关键性质。

[5] arXiv:2509.23701 (cross-list from math.FA) [cn-pdf, pdf, html, other]

Title: Positive contractive projections in Schatten Spaces Show Chinese title

Title: Schatten空间中的正则压缩投影

Estelle Boffy

Subjects: Functional Analysis (math.FA) ; Operator Algebras (math.OA)

We characterize the positively 1-complemented subspaces of $S^p$, for $1\leq p<\infty$, where $S^p$ denotes the Schatten spaces. Building on the work of Arazy and Friedman, who described the 1-complemented subspaces of $S^p$, for $1\leq p\neq 2 <\infty$, we establish that there are five mutually distinct types of indecomposable positively 1-complemented subspaces in $S^p$. Moreover, every positively 1-complemented subspace of $S^p$ can be expressed as a direct sum of some of these indecomposable subspaces. Show Chinese abstract

我们表征了$S^p$的正 1-补子空间，其中$1\leq p<\infty$，这里$S^p$表示 Schatten 空间。 在 Arazy 和 Friedman 的工作基础上，他们描述了$S^p$的 1-补子空间，对于$1\leq p\neq 2 <\infty$，我们确定在$S^p$中存在五种相互不同的不可分解的正 1-补子空间类型。 此外，$S^p$的每个正 1-补子空间都可以表示为这些不可分解子空间的直和。

[6] arXiv:2509.23950 (cross-list from math.KT) [cn-pdf, pdf, html, other]

Title: On Chern classes of almost representations Show Chinese title

Title: 论几乎表示的陈类

Marius Dadarlat, Forrest Glebe

Subjects: K-Theory and Homology (math.KT) ; Group Theory (math.GR) ; Operator Algebras (math.OA)

For a discrete group $\Gamma$, we study vector bundles $E_\rho$ on compact subsets of $B\Gamma$ associated to almost representations $\rho:\Gamma \to U(n)$. We compute the first Chern class of $E_\rho$ in terms of $\rho$. When $\rho$ is both projective and almost multiplicative, we determine its Chern character. These invariants yield obstructions to perturbing almost representations to those arising from projective representations. For residually finite amenable groups, the $K$-theory classes of $E_\rho$ classify almost representations up to stable equivalence. Finally, for $\mathbb{Z}^d$, $\mathbb{Z}\times \mathbb{H}_3$, and $\mathbb{H}_3\times \mathbb{H}_3$, we construct explicit almost representations with prescribed Chern classes. Show Chinese abstract

对于离散群$\Gamma$，我们研究在$B\Gamma$的紧子集上与几乎表示$\rho:\Gamma \to U(n)$关联的向量丛$E_\rho$。 我们以$\rho$的形式计算$E_\rho$的第一陈类。 当$\rho$既是投影的又是几乎乘法的时候，我们确定其陈特征。 这些不变量给出了将几乎表示扰动为来自投影表示的障碍。 对于剩余有限的可解群，$K$-理论类别的$E_\rho$分类几乎表示直到稳定等价。 最后，对于$\mathbb{Z}^d$，$\mathbb{Z}\times \mathbb{H}_3$和$\mathbb{H}_3\times \mathbb{H}_3$，我们构造具有预定陈类的显式几乎表示。

[7] arXiv:2509.24106 (cross-list from math.FA) [cn-pdf, pdf, html, other]

Title: Twisted crossed products of Banach algebras Show Chinese title

Title: 巴拿赫代数的扭曲交叉乘积

Alonso Delfín, Carla Farsi, Judith Packer

Comments: AMSLaTeX; 27 pages

Subjects: Functional Analysis (math.FA) ; Dynamical Systems (math.DS) ; Operator Algebras (math.OA)

Given a locally compact group $G$, a nondegenerate Banach algebra $A$ with a contractive approximate identity, a twisted action $(\alpha, \sigma)$ of $G$ on $A$, and a family $\mathcal{R}$ of uniformly bounded representations of $A$ on Banach spaces, we define the twisted crossed product $F_\mathcal{R}(G,A,\alpha, \sigma)$. When $\mathcal{R}$ consists of contractive representations, we show that $F_\mathcal{R}(G,A,\alpha, \sigma)$ is a Banach algebra with a contractive approximate identity, which can also be characterized by an isometric universal property. As an application, we specialize to the $L^p$-operator algebra setting, defining both the $L^p$-twisted crossed product and the reduced version. Finally, we give a generalization of the so-called Packer-Raeburn trick to the $L^p$-setting, showing that any $L^p$-twisted crossed product is "stably" isometrically isomorphic to an untwisted one. Show Chinese abstract

给定一个局部紧群$G$，一个具有压缩逼近单位的非退化巴拿赫代数$A$，一个由$G$在$A$上的扭曲作用$(\alpha, \sigma)$，以及一个由$A$在巴拿赫空间上的统一有界表示族$\mathcal{R}$，我们定义扭曲的交叉积$F_\mathcal{R}(G,A,\alpha, \sigma)$。 当$\mathcal{R}$包含压缩表示时，我们证明$F_\mathcal{R}(G,A,\alpha, \sigma)$是一个具有压缩近似单位的巴拿赫代数，也可以通过一个等距的普遍性质来表征。 作为应用，我们专门考虑$L^p$算子代数的设定，定义了$L^p$-扭曲交叉积以及其约化版本。 最后，我们将所谓的 Packer-Raeburn 技巧推广到$L^p$的设定中，表明任何$L^p$-扭曲交叉积都“稳定地”等距同构于一个无扭曲的交叉积。

Replacement submissions (showing 5 of 5 entries )

[8] arXiv:2502.15362 (replaced) [cn-pdf, pdf, html, other]

Title: $C^*$-extreme points of unital completely positive maps on real $C^*$-algebras Show Chinese title

Title: $C^*$-实$C^*$-代数上单位完全正映射的极点

Anand O. R, K. Sumesh, Arindam Sutradhar

Comments: A shorter proof of Theorem 3.18 is added. Proposition 4.3 and Lemma 4.4 are newly added. Modified the proof of Proposition 4.8. This work is accepted for publication in Studia Mathematica

Subjects: Operator Algebras (math.OA) ; Functional Analysis (math.FA)

In this paper, we investigate the general properties and structure of $C^*$-extreme points within the $C^*$-convex set $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$ of all unital completely positive (UCP) maps from a unital real $C^*$-algebra $\mathcal{A}$ to the algebra $B(\mathcal{H})$ of all bounded real linear maps on a real Hilbert space $\mathcal{H}$. We analyze the differences in the structure of $C^*$-extreme points between the real and complex $C^*$-algebra cases. In particular, we show that the necessary and sufficient conditions for a UCP map between matrix algebras to be a $C^*$-extreme point are identical in both the real and complex matrix algebra cases. We also observe significant differences in the structure of $C^*$-extreme points when $\mathcal{A}$ is a commutative real $C^*$-algebra compared to when $\mathcal{A}$ is a commutative complex $C^*$-algebra. We provide a complete classification of the $C^*$-extreme points of $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$, where $\mathcal{A}$ is a unital commutative real $C^*$-algebra and $\mathcal{H}$ is a finite-dimensional real Hilbert space. As an application, we classify all $C^*$-extreme points in the $C^*$-convex set of all contractive skew-symmetric real matrices in $M_n(\mathbb{R})$. Show Chinese abstract

在本文中，我们研究了在实单位 $C^*$-代数 $\mathcal{A}$到实希尔伯特空间 $\mathcal{H}$上所有有界实线性映射的代数 $B(\mathcal{H})$中的所有单位完全正（UCP）映射的实单位 $C^*$-凸集 $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$内的 $C^*$-极点的一般性质和结构。 我们分析实数和复数$C^*$-代数情况下$C^*$-极端点结构的差异。 特别是，我们证明在矩阵代数之间的UCP映射成为$C^*$-极端点的充要条件在实数和复数矩阵代数情况下是相同的。 我们还观察到，当$\mathcal{A}$是一个交换的实$C^*$-代数时，$C^*$极端点的结构与当$\mathcal{A}$是一个交换的复$C^*$-代数时存在显著差异。 We provide a complete classification of the $C^*$-extreme points of $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$, where $\mathcal{A}$ is a unital commutative real $C^*$-algebra and $\mathcal{H}$ is a finite-dimensional real Hilbert space. 作为应用，我们分类了所有在$C^*$-极值点的$C^*$-凸集中所有收缩反对称实矩阵的$M_n(\mathbb{R})$。

[9] arXiv:2506.19940 (replaced) [cn-pdf, pdf, other]

Title: Strong convergence to operator-valued semicirculars Show Chinese title

Title: 到算子值半圆形的强收敛

David Jekel, Yoonkyeong Lee, Brent Nelson, Jennifer Pi

Comments: 37 pages; updated article includes additional applications

Subjects: Operator Algebras (math.OA) ; Probability (math.PR)

We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices $\eta = (\eta_{i,j})_{i,j \in I}$. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of $\eta_{i,j}$, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a $B$-valued semicircular family and generators of the base algebra $B$. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors. Show Chinese abstract

我们建立了一个框架，用于矩阵模型弱收敛和强收敛到由算子值协方差矩阵$\eta = (\eta_{i,j})_{i,j \in I}$参数化的算子值半圆系统。非交换多项式被可以涉及迭代应用$\eta_{i,j}$的协方差多项式所替代，从而得到协方差定律的概念。我们给出了一般高斯随机矩阵和确定性矩阵弱收敛和强收敛到$B$值半圆族和基代数$B$生成元的充分条件。特别是，我们得到了连续加权高斯 Wigner 矩阵的算子值强收敛，例如具有连续截断的高斯带状矩阵，并且我们为插值自由群因子构造了自然的强收敛矩阵模型。

[10] arXiv:2408.10232 (replaced) [cn-pdf, pdf, html, other]

Title: Theory of $q$-commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality Show Chinese title

Title: $q$-交换收缩的理论-II：正则扩张，Brehmer的正性与冯·诺伊曼不等式

Sourav Pal, Prajakta Sahasrabuddhe, Nitin Tomar

Comments: 30 pages, Submitted to journal

Subjects: Functional Analysis (math.FA) ; Operator Algebras (math.OA)

It is well-known that a commuting family of contractions possesses a regular unitary dilation if and only if it satisfies Brehmer's positivity condition. We extend this theorem to any family $\mathcal T$ of $q$-commuting contractions with $\|q\|=1$ by showing the equivalence of the following three statements: $(i)$ $\mathcal T$ admits a regular $q$-unitary dilation; $(ii)$ $\mathcal T$ satisfies Brehmer's positivity condition; $(iii)$ $\mathcal T$ admits a $Q$-unitary dilation for a family of $Q$-commuting unitaries. We achieve the first part of the result by an application of Stinespring's dilation theorem on a particular completely positive map acting on a quotient algebra of a group $C^*$-algebra, where the underlying group is a free group, and the second part is obtained by an application of Naimark's theorem. Next, we find several cases when $\mathcal{T}$ admits a regular $q$-unitary dilation and establish a von Neumann type inequality for such a $q$-commuting family. Show Chinese abstract

众所周知，一个交换的压缩算子族具有正则的单位扩张当且仅当它满足Brehmer的正性条件。 我们将这个定理推广到任何族$\mathcal T$的$q$交换压缩算子，其中$\|q\|=1$，通过证明以下三个陈述的等价性：$(i)$ $\mathcal T$ 具有一个正则的$q$-酉扩张；$(ii)$ $\mathcal T$ 满足 Brehmer 的正性条件；$(iii)$ $\mathcal T$ 对于一个$Q$交换的酉算子族具有一个$Q$-酉扩张。 我们通过在特定完全正映射上应用Stinespring的扩张定理来实现结果的第一部分，该映射作用于一个群的商代数，其中基础群是一个自由群，第二部分则是通过应用Naimark定理得到的。 接下来，我们找到几种当$C^*$时$\mathcal{T}$允许一个正则$q$-酉扩张的情况，并为这样的$q$-交换族建立一个冯·诺依曼型不等式。

[11] arXiv:2412.08130 (replaced) [cn-pdf, pdf, html, other]

Title: A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras Show Chinese title

Title: 一种非标准分析方法用于限制算子和Roe-like代数中的Fredholm性

Liang Guo, Jin Qian, Qin Wang

Comments: Totally rewritten in a more nonstandard analysis flavor

Subjects: Functional Analysis (math.FA) ; Metric Geometry (math.MG) ; Operator Algebras (math.OA)

Let $(X,d)$ be a uniformly locally finite metric space, and $T$ an operator in the uniform Roe algebra $C_u^*(X)$ (or uniform quasi-local algebra $C_{ql}^*(X)$). In this paper, we introduce the concept of limit operators of $T$ on galaxies in the nonstandard extension of $X$, and prove that $T$ is a generalized Fredholm operator with respect to the ghost ideal in $C_u^*(X)$ (or $C_{ql}^*(X)$) if and only if all limit operators on afar galaxies are invertible, and their inverses are uniformly bounded. In particular, if $X$ has Yu's Property A, then $T$ is a Fredholm operator if and only if all limit operators on afar galaxies are invertible. Using techniques in nonstandard analysis, our result strengthens a work of \v{S}pakula--Willett \cite{SpW} on the characterization of Fredholmness by using less limit operators. Show Chinese abstract

设$(X,d)$为一个均匀局部有限的度量空间，$T$为均匀Roe代数$C_u^*(X)$（或均匀准局部代数$C_{ql}^*(X)$）中的一个算子。 In this paper, we introduce the concept of limit operators of $T$ on galaxies in the nonstandard extension of $X$, and prove that $T$ is a generalized Fredholm operator with respect to the ghost ideal in $C_u^*(X)$ (or $C_{ql}^*(X)$) if and only if all limit operators on afar galaxies are invertible, and their inverses are uniformly bounded. In particular, if $X$ has Yu's Property A, then $T$ is a Fredholm operator if and only if all limit operators on afar galaxies are invertible. 使用非标准分析中的技术，我们的结果加强了Špakula--Willett \cite{SpW} 关于通过较少极限算子来表征弗雷德霍姆性的研究。

[12] arXiv:2504.09576 (replaced) [cn-pdf, pdf, html, other]

Title: Bimodule Quantum Markov Semigroups Show Chinese title

Title: 双模量子马尔可夫半群

Jinsong Wu, Zishuo Zhao

Subjects: Quantum Physics (quant-ph) ; Functional Analysis (math.FA) ; Operator Algebras (math.OA)

We present a systematic investigation of bimodule quantum Markov semigroups within the framework of quantum Fourier analysis. Building on the structure of quantum symmetries, we introduce the concepts of bimodule equilibrium and bimodule detailed balance conditions, which not only generalize the classical notions of equilibrium and detailed balance but also expose interesting structures of quantum channels. We demonstrate that the evolution of densities governed by the bimodule quantum Markov semigroup is the bimodule gradient flow for the relative entropy with respect to quantum symmetries. Consequently, we obtain bimodule logarithmic Sobelov inequalities and bimodule Talagrand inequality with respect to a hidden density from higher dimensional structure. Furthermore, we establish a bimodule Poincar\'{e} inequality for irreducible inclusions and relative ergodic bimodule quantum semigroups. Show Chinese abstract

我们对量子傅里叶分析框架内的双模态量子马尔可夫半群进行了系统研究。 在量子对称性的结构基础上，我们引入了双模态平衡和双模态细致平衡条件，这些不仅推广了经典的平衡和细致平衡概念，还揭示了量子通道有趣的结构。 我们证明了由双模态量子马尔可夫半群支配的密度演化是相对于量子对称性的相对熵的双模态梯度流。 因此，我们从高维结构中得到了关于隐藏密度的双模态对数索博列夫不等式和双模态塔拉甘德不等式。 此外，我们为不可约包含和相对遍历双模态量子半群建立了双模态泊松不等式。