Morita equivalence of smooth noncommutative tori
George A. Elliott, Hanfeng Li
我们证明了在一般情况下，两个n × n实反对称矩阵所对应的光滑非交换环面是Morita等价的当且仅当这两个矩阵在自然SO（n，n）的同一轨道上|Z）动作。
We show that in the generic case the smooth noncommutative tori associated to two n by n real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO(n, n| Z) action.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Strong Morita equivalence of higher-dimensional noncommutative tori
我们证明了SO（n，n）的同一轨道上的矩阵|Z）作用在n个\times n个斜对称矩阵空间上，在C* 代数和光滑代数水平上给出了强Morita等价的非交换环面.这证明了Rieffel和Schwarz的一个猜想。
Strong Morita equivalence of higher-dimensional noncommutative tori. II
我们证明了两个C*-代数非交换环面是强Morita等价的当且仅当它们有同构的序K_0-群和中心，扩张N。C. Phillips在单代数情形下的结果。这一结果也推广到了任意n-生成的阿贝尔群的扭群C* 代数。
Morita equivalence of multidimensional noncommutative tori
人们可以用反对称的$n\times n$ -矩阵$θ$来描述$n$维非交换环面.我们构建了一个行动的组$SO(n,n|\bf Z)$的空间上的反对称矩阵，并表明，一般来说，矩阵属于同一轨道的这个群体给森田等价环面。简述了它在物理学中的一些应用。
Quantization and Morita equivalence for constant Dirac structures on tori
定义了环面上常数Dirac结构的C*-代数量子化，推广了Poisson结构的标准量子化.我们证明了在O（n，n）的自然作用的同一轨道上的Dirac结构|Z）给出了Morita等价代数，完善和推广了Rieffel和Schwarz的一个定理.
Morita Equivalence and Spectral Triples on Noncommutative Orbifolds
设$G$是一个有限群.研究了酉$G$ -代数的非交换几何.几何结构由与群作用相关的交叉积代数上的谱三元组确定。这种结构被看作是非对易orbifold的代表。在研究经典轨道群胚的基础上，建立了交叉积谱三元组的Morita等价。非对易轨道折叠是交叉积谱三元组的Morita等价类。作为这个森田理论的一个特例，人们可以在非对易水平上研究$G$ -作用的自由性。在自由作用量的情况下，交叉积形式化约为$G$ -不变函数代数上通常的谱三元组形式。
Logarithmic Sobolev Inequalities of Fractional Order on Noncommutative Tori
本文证明了任意维度$n\geq 2$的非交换$n$-环面上的分数阶对数Sobolev不等式。
Morita equivalence and duality
Connes，Douglas，Schwarz[1]证明了非对易环面上的M（atrix）理论可以紧化.证明了Morita等价环面上的紧化是物理等价的。这一陈述可以看作是文[1]中的非经典对偶在二维环面上的推广。
Heisenberg-Weyl Representations and Morita equivalence for crossed products of Noncommutative solenoids
研究非交换螺线管交叉积的强莫里塔等价，构造不变乘子，建立海森堡 - 外尔框架，通过酉算子扩展等价双模，得到无限和有限循环子群交叉积的强莫里塔等价。
Any finite algebraic Galois covering corresponds to an algebraic Morita equivalence. Here the $C^*$-algebraic analog of this fact is proven, i.e. any noncommutative finite-fold covering corresponds to a strong Morita equivalence.
A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this paper, we extend recent results on the modular geometry on noncommutative two tori to a larger class of noncommutative manifolds: toric noncommutative manifolds. The first contribution of this work is a pseudo differential calculus which is suitable for spectral geometry on toric noncommutativ…
The Gauss-Manin connection and noncommutative tori
利用Getzler的Gauss-Manin联络证明了非对易环面光滑变形的周期循环上同调的不变性。我们明确地计算了平行移动映射，并使用它们来描述这种变形下的Chern-Connes配对的行为。
The curvature of the noncommutative torus $T^2_θ$ ($θ$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_θ= C(T^2_θ)$ with other toric algebras $A_{θ'}$. In the enlarged context the conformal metric on $T^2_θ$ is exchanged with an arbitrary Hermitian metric…
A Noncommutative Residue on Tori and a Semiclassical Limit
本文定义了任意维环面上经典欧氏伪微分算子的一个非交换剩余。我们证明了，直到乘以一个常数，它是唯一的迹代数的经典伪微分算子模无限光滑算子。在两个环面的情况下，我们证明了非交换剩余是定义在非交换两个环面上的经典伪微分算子上的非交换剩余的半经典极限。
We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the $K$-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an…
Noncommutative protori and inductive spectral triples
研究高维非交换环面的归纳极限（称为非交换原环），通过计算Elliott不变量并构造相容的谱三元组，给出了分类准则。
Appendix to V. Mathai and J. Rosenberg's paper "A noncommutative sigma-model"
证明了Rosenberg关于二维非对易环面上酉元能量的极小值的一个猜想，并回答了他关于二维非对易环面上酉 *-自同态能量的下界的一个问题.
The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure
本文对具有任意平移不变复结构的非交换两环面$\mathbb{T}_θ^2$给出了Connes和Tretkoff的Gauss-Bonnet定理的证明。更准确地说，我们表明，对于任何复数$τ$在上半平面，代表共形类的度量$\mathbb{T}_θ^2$，和Weyl因子给出的一个积极的可逆元素$k \in C^{\infty}(\mathbb{T}_θ^2)$，在原点的值，$ζ(0)$，光谱zeta函数的拉普拉斯$\triangle'$附加到$(\mathbb{T}_θ^2, τ, k)$是独立的$τ$和$k$。
In this paper we establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators.The latter is the unique trace up to constant multiple on non-integer order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and Lévy-Neira-Paycha.
Inverse-Closed Subalgebras of Noncommutative Tori
本文给出了一般高维非交换环面上的逆闭（Banach）子代数的系统构造
On Noncommutative Geometry of Orbifolds
Orbifold是真Lie群胚的Morita等价类.光滑不变函数代数上的谱三元组的酉等价类与任何紧自旋轨道相关联。在一个有效的自旋轨道的情况下，我们构建了一个收集的谱三倍的光滑卷积代数的代表的Morita等价类。
Spectral Triples on noncommutative solenoids from the standard spectral triples on quantum tori
研究非交换索引作为量子紧致空间的诱导极限，探讨标准量子紧致空间谱三元组是否收敛到非交换索引上的谱三元组，通过适当扰动构造此类谱三元组。
We prove a noncompact version of Haagerup and Rørdam's result about continuous paths of the rotation $C^*$-algebras. It gives a continuous Moyal deformation of Euclidean plane. Moveover, the construction is generalized to noncommutative Euclidean spaces of dimension $d\ge 2$. As a corollary, we obtain Lip$^{\frac12}$ continuous maps for the generators of noncommutative $d$-tori.
Certain actions of finitely generated groups on higher dimensional noncommutative tori
本文研究了高维非交换环面上的非正则生成阿贝尔群的某些作用。给定一个维数$d$和一个非交换生成的阿贝尔群$G$，我们利用一个函数来判定是否存在一个简单的非交换$d$ -环面，它允许$G$的一个特定作用.我们还描述了$G$的条件下，相关的交叉积是AF代数。
Gabor Analysis, Noncommutative Tori and Feichtinger's algebra
我们指出了Gabor分析和非交换分析之间的联系。特别是，强森田等价的非交换环面出现作为基本的设置为Gabor分析，因为建设的等价双模的非交换环面有一个自然的公式在Gabor分析的概念。作为应用，我们证明了Feichtinger代数是这样一个等价双模.此外，我们提出了Connes的投射模的非交换环面的建设和相关的广义维纳引理扭曲卷积的Gröchenig和Leinert。最后，基于Morita等价性的结果，给出了关于Gabor框架算子对偶原子存在性的Wexler-Raz双正交关系的一种方法。
Using Rieffel's construction of projective modules over higher dimensional noncommutative tori, we construct projective modules over some continuous field of C*-algebras whose fibers are noncommutative tori. Using a result of Echterhoff et al., our construction gives generators of $K_0$ of all noncommutative tori.
Artin $L$-functions and noncommutative tori
本文利用Deninger思想证明Artin $L$-函数与非交换环面的函数一致，为非交换环面建立了Langlands互反律。
An analogue of Gross' logarithmic Sobolev inequality for a class of elements of noncommutative two tori is proved.
On complex and noncommutative tori
研究复代数曲线的“非交换几何“。作为第一步，我们阐明了椭圆曲线或复环面与C*-代数T_t={u，v| vu=exp（2πit）uv}，或非交换环面。主要结果是在态射下同构的椭圆曲线映射到Morita等价的非交换环面。我们的方法是基于黎曼曲面的长度谱的刚性。
We revisit and generalize the application of a method introduced by Latrémolière and Packer for constructing finitely generated projective modules over the noncommutative solenoid C*-algebras. By realizing them as direct limits of rotation algebras, the method constructs directed systems of equivalence bimodules between rotation algebras that satisfy the necessary compatibility conditions to build Morita equivalence bimodules between the direct limit C*-algebras. In the irrational case, we use a fixed projection in a matrix algebra over the rotation algebra satisfying a key condition to build …
Quantum symmetries of noncommutative tori
本文提出利用有限维数据在归纳极限 C*-代数上构造融合范畴作用的方法，并据此在不同维数的非交换环面上实现 Haagerup–Izuma、$E_8$ 子因子偶数部分及 $\ ext{PSU}(2)_{15}$ 的非可逆量子对称性。
Langlands reciprocity for the even dimensional noncommutative tori
我们推测了高维狄利克雷特征标的一个显式公式;该公式基于所谓非交换环面的K理论。它被证明，我们的猜想是真实的二维和一维（退化）的非交换环面;在第二种情况下，人们得到一个非交换模拟的阿廷互易定律。
We point out a link between the theorem of Balian and Low on the non-existence of well-localized Gabor-Riesz bases and a constant curvature connection on projective modules over noncommutative tori.
In this paper, we establish the full $L_p$ boundedness of noncommutative Bochner-Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in \cite{CXY13} in the sense of the $L_p$ convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner-Riesz means.
Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori
我们证明了非对易黎曼流形的Weyl定律的类似物，即非对易两环面$\mathbb{T}_θ^2$，它具有一般的平移不变共形结构和Weyl共形因子.这是通过研究扰动拉普拉斯算子在$\mathbb{T}_θ^2$上的特征值的渐近分布来实现的。通过证明$\mathbb{T}_θ^2$上的-2阶伪微分算子的Dixontium迹与非交换剩余重合，证明了Connes迹定理的类似性.
Scalar Curvature for Noncommutative Four-Tori
本文研究了非对易4-环面$\mathbb{T}_θ^4$的曲面几何。我们使用一个外尔共形因子扰动标准体积形式，并获得拉普拉斯编码的局部几何信息。我们使用Connes' pseudodifferential微积分显式计算的条款在小的时间热核展开的扰动拉普拉斯对应的体积和数量曲率的$\mathbb{T}_θ^4$。我们建立了类似的Weyl定律，定义了一个非交换剩余，证明了类似的Connes迹定理，并找到了明确的公式，描述的数量曲率的$\mathbb{T}_θ^4$的本地功能。我们还研究了这些空间的爱因斯坦-希尔伯特作用的模拟，并表明具有常数数量曲率的度量是这个行动的关键。
This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes' pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a detailed description is still missing. In this paper, we focus on constructing an oscillating integral for noncommutative tori and laying down the main functional analysis ground for understanding Connes' pseudodifferential calculus. In particular, this allows us to give a precise explanation of the definition of pseudodifferential operators on noncommutative tori…
In this article we describe extensions of some K-theory classes of Heisenberg modules over higher-dimensional noncommutative tori to projective modules over crossed products of noncommutative tori by finite cyclic groups, aka noncommutative orbifolds. The two dimensional case was treated by Echterhoff, Lück, Phillips and Walters. Our approach is based on the theory of metaplectic transformations of the representation theory of the Heisenberg group. We also describe the generators of the K-groups of the crossed products of flip actions by $\mathbb{Z}_2$ on 3-dimensional noncommutative tori.
The Riemann-Roch Theorem on higher dimensional complex noncommutative tori
本文证明了配备复结构的任意维非交换环面上的黎曼-罗赫定理与霍奇定理的类似物，并探讨了区分非交换阿贝尔簇与非代数非交换复环面的理论意义。
Projections in noncommutative tori and Gabor frames
我们描述了两个看似不同的问题之间的联系：（a）在非交换环面的投影的建设，（b）紧的Gabor框架的建设。本研究依赖于非交换环面上的投射模的Gabor分析的解释。主要结果表明，Rieffel关于非交换环面上投影存在的条件等价于紧Gabor框架的Wexler-Raz双正交关系。因此，我们能够调用的结果存在的Gabor框架在建设的非交换环面的投影。特别是，与高斯生成的Gabor框架相关的投影被证明是博卡投影。我们的方法，博卡的投影，使我们能够表征的范围存在的博卡的投影。我们的主要结果的介绍提供了一个自然的方法，Wexler-Raz双正交关系的Hilbert C*-模在非交换环上。
On a desingularization of the moduli space of noncommutative tori
证明了非对易环面A的模空间可被群Ext（A，A）自然去奇异化.也就是说，我们证明了对（A，Ext（A，A））的模空间同胚于一个穿孔的二维球面。证明是基于一个对应关系（共变函子）之间的复杂和非交换环面。
A Noncommutative Sigma Model
我们用非对易环面取代了经典的弦理论中的参数空间和世界时的概念，并考虑了这些空间之间的映射。研究了不同非对易环面之间映射的动力学，得到了Polyakov作用的非对易推广。详细研究了量子环面以及不同量子环面之间的 *-同态。研究了量子环面的有限维表示，计算了量子环面的分拆函数和其他路径积分。最后证明了不同非交换环之间映射的存在性定理。
This paper is the 2nd part of a two-paper series whose aim is to give a detailed description of Connes' pseudodifferential calculus on noncommutative $n$-tori, $n\geq 2$. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension $n\geq 2$. This includes the main results mentioned in the original notes of Connes and Baaj. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negativ…
$L$-function of noncommutative tori
本文引入非交换环面的$L$-函数类似物，证明其等同于具有复乘的椭圆曲线的Hasse-Weil $L$-函数，并得出实乘非交换环面的局部化公式。
In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.
In this paper, we introduce a parametric pseudodifferential calculus on noncommutative $n$-tori which is a natural nest for resolvents of elliptic pseudodifferential operators. Unlike in some previous approaches to parametric pseudodifferential calculi, our parametric pseudodifferential calculus contains resolvents of elliptic pseudodifferential operators that need not be differential operators. As an application we show that complex powers of positive elliptic pseudodifferential operators on noncommutative $n$-tori are pseudodifferential operators. This confirms a claim of Fathi-Ghorbanpour-K…
It is proved that a maximal abelian subalgebra of the noncommutative torus commutes with the Laplace operator on a complex torus. As a corollary, one gets an analog of the Poisson summation formula for noncommutative tori.
Noncommutative Coverings of Quantum Tori
本文提出了一种非交换空间覆盖的框架，并研究了无理量子环的非交换覆盖特性。
We consider a family of higher-dimensional noncommutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori, and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz noncommutative solenoids are direct limits of the Toeplitz extensions of noncommutative tori. We consider natural dynamics on these Toeplitz algebras, and compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrised by the probability measures on an (ordinary) solenoid.
Non-abelian class field theory and higher dimensional noncommutative tori
本研究探究Drinfeld模与偶数维非交换环面的关联，以此为基础构建非阿贝尔类域论，还构造出伽罗瓦扩张的显式生成元，推进了相关数论领域的研究。
In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$n\geq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metr…
Pseudodifferential Operators on Noncommutative Tori: a Survey
本文综述非交换环面上伪微分算子的多种构造方法，包括动力系统、傅里叶级数类比、Rieffel形变及谱三元组关联。
Drinfeld modules as noncommutative tori
本文利用 Drinfeld 模与非交换环面的相似性，尝试为数域构建显式类域论，并重点详析虚二次数域情形。
Modular Curvature for Noncommutative Two-Tori
本文在与扰动Dolbeault算子相关的谱三重泛函的框架下，研究了非交换2-环面上平坦度量的非交换Weyl因子的共形形变的曲率.高斯曲率的模拟结果是对应于由保形因子定义的非迹权的模算子中的两个函数的和，应用于涉及相同因子的导数的表达式。第一个是伯努利数的生成函数，并应用于保形因子的非交换拉普拉斯算子，而第二个是二元函数，并应用于因子的一阶导数中的二次型。进一步的成果的文件包括变分证明的高斯-邦纳定理的非交换2-环面，模块模拟Polyakov的共形异常公式的正则化决定因素的拉普拉斯算子，概念上的理解模块曲率梯度的Ray-Singer分析扭转，并证明使用运营商的积极性，规模不变的版本，后者假设其极值只有在平坦的度量。
In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension~$n\geq 2$. We use them to derive Cwikel-Lieb-Rozenblum inequalities and and Lieb-Thirring inequalities for the number of negative eigenvalues of fractional Schroedinger operators on noncommutative tori in any dimension~$n\geq 2$. The latter leads to a Sobolev inequality for noncommutative tori. On the way we establish a "borderline version" of the abstract Birman-Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator wi…
Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces
本文证明了Dabrowski和Sitarz引入的弯曲非交换环面是Leibniz量子紧度量空间，并且当这个群被赋予一个自然长度函数时，它们在正则表示中的量子环面的交换子中的可逆矩阵群上形成一个连续族.
Levi-Civita's Theorem for Noncommutative Tori
我们将展示如何定义黎曼度量和连接上的非交换环面的方式，一个类似的列维-奇维塔定理的存在性和唯一性的黎曼连接举行。主要的新颖之处在于我们需要使用两种不同的非对易向量场概念。Levi-Civita定理使得使用通常的公式来定义黎曼曲率成为可能。
Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces
在本研究中，我们将非交换环面上的非交换几何与Gabor分析联系起来，其中第一种方法的根源是算子代数，第二种方法的根源是时频分析。因此，我们能够调用现代算子代数的方法，例如巴拿赫代数的拓扑稳定秩，以利用更深层次的性质的Gabor框架。此外，我们能够推广的结果，由于康纳斯和Rieffel的投射模上的非交换环面的巴拿赫代数，这出现在一个自然的方式在Gabor分析。本研究的主要目标是双重的：（i）在非交换环面上的投射模的Gabor分析的解释和（ii）的Morita-Rieffel非交换环面之间的等价性是自然的框架的Gabor框架的对偶理论。更具体地说，我们解释的非交换环面上的投射模的生成元调制空间的多窗口的Gabor框架的Gabor原子。此外，我们表明，这意味着{ \it存在}的好的多窗口Gabor框架的调制空间的Gabor原子在Feichtinger的代数和施瓦兹空间。
The C*-algebra generated by the left-regular representation of $\mathbb{N}^n$ twisted by a $2$-cocycle is a Toeplitz extension of an $n$-dimensional noncommutative torus, on which each vector $r \in [0,\infty)^n$ determines a one-parameter subgroup of the gauge action. We show that the equilibrium states of the resulting C*-dynamical system are parametrised by tracial states of the noncommutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of $r$. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain de…
Schatten classes on noncommutative tori: Kernel conditions
本文研究了量子环面上积分算子的Schatten类隶属问题，通过引入非交换积分核准则并结合非交换Schwartz核定理，建立了判定有界算子Schatten类性质的具体测试方法。
We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。