George A. Elliott, Hanfeng Li
We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K_0-groups and centers, extending N. C. Phillips's result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups.
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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.
We show that matrices in the same orbit of the SO(n, n|Z) action on the space of n\times n skew-symmetric matrices give strong Morita equivalent noncommutative tori, both at the C*-algebra level and at the smooth algebra level. This proves a conjecture of Rieffel and Schwarz.
One can describe an $n$-dimensional noncommutative torus by means of an antisymmetric $n\times n$-matrix $θ$. We construct an action of the group $SO(n,n|\bf Z)$ on the space of antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched.
We define a C*-algebraic quantization of constant Dirac structures on tori, which extends the standard quantization of Poisson structures. We prove that Dirac structures in the same orbit of a natural action of O(n,n|Z) give rise to Morita equivalent algebras, completing and extending a theorem of Rieffel and Schwarz.
Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the $G$-action on the …
We use Getzler's Gauss-Manin connection to prove the invariance of periodic cyclic cohomology for the smooth deformation of noncommutative tori. We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern-Connes pairing under this deformation.
It was shown by Connes, Douglas, Schwarz[1] that one can compactify M(atrix) theory on noncommutative torus. We prove that compactifications on Morita equivalent tori are physically equivalent. This statement can be considered as a generalization of non-classical duality conjectured in [1] for two-dimensional tori.
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.
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。
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。
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.
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 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。
For an action of a finite cyclic group $F$ on an $n$-dimensional noncommutative torus $A_θ,$ we give sufficient conditions when the fundamental projective modules over $A_θ$, which determine the range of the canonical trace on $A_θ,$ extend to projective modules over the crossed product C*-algebra $A_θ\rtimes F.$ Our results allow us to understand the range of the canonical trace on $A_θ\rtimes F$, and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of $\mathbb{Z}_2$ on any $n$-dimens…
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…
We define a noncommutative residue for classical Euclidean pseudodifferential operators on a torus of arbitrary dimension. We prove that, up to multiplication by a constant, it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators. In the case of the two torus, we show that the noncommutative residue is the semiclassical limit of a noncommutative residue defined on classical pseudodifferential operators on noncommutative two tori.
An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the smooth convolution algebras of the representatives of the Morita equivalence class.
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。
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.
We give a systematic construction of inverse-closed (Banach) subalgebras in general higher-dimensional non-commutative tori
We point out a connection between Gabor analysis and noncommutative analysis. Especially, the strong Morita equivalence of noncommutative tori appears as underlying setting for Gabor analysis, since the construction of equivalence bimodules for noncommutative tori has a natural formulation in the notions of Gabor analysis. As an application we show that Feichtinger's algebra is such an equivalence bimodule. Furthermore, we present Connes's construction of projective modules for noncommutative tori and the relevance of a generalization of Wiener's lemma for twisted convolution by Gröchenig and …
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 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…
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。
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.
The "noncommutative geometry" of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C*-algebras T_t={u,v | vu=exp(2πit)uv}, or noncommutative tori. The main result says that under the morphism isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces.
We conjecture an explicit formula for the higher dimensional Dirichlet character; the formula is based on the K-theory of the so-called noncommutative tori. It is proved, that our conjecture is true for the two-dimensional and one-dimensional (degenerate) noncommutative tori; in the second case, one gets a noncommutative analog of the Artin reciprocity law.
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.
We study certain actions of finitely generated abelian groups on higher dimensional noncommutative tori. Given a dimension $d$ and a finitely generated abelian group $G$, we apply a certain function to detect whether there is a simple noncommutative $d$-torus which admits a specific action by $G$. We also describe the condition of $G$ under which the associated crossed product is an AF algebra.
We prove a conjecture of Rosenberg about the minimal value for energies of untaries in the two-dimensional noncommutative tori and answer a question of his about lower bounds for energies of unital *-endomorphisms of the two-dimensional noncommutative tori.
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。
In this paper we give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori $\mathbb{T}_θ^2$ equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number $τ$ in the upper half plane, representing the conformal class of a metric on $\mathbb{T}_θ^2$, and a Weyl factor given by a positive invertible element $k \in C^{\infty}(\mathbb{T}_θ^2)$, the value at the origin, $ζ(0)$, of the spectral zeta function of the Laplacian $\triangle'$ attached to $(\mathbb{T}_θ^2, τ, k)$ is independent of $τ$ and $k$.
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。
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.
We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori, (b) the construction of tight Gabor frames. The present investigation relies an interpretation of projective modules over noncommutative tori in terms of Gabor analysis. The main result demonstrates that Rieffel's condition on the existence of projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relations for tight Gabor frames. Therefore we are able to invoke results on the existence of Gabor frames in the construction of projections in n…
It is shown that the moduli space of the noncommutative tori A admits a natural desingularization by the group Ext (A, A). Namely, we prove that the moduli space of pairs (A, Ext (A,A)) is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.
$L$-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.
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 …
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.
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…
We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $\mathbb{T}_θ^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on $\mathbb{T}_θ^2$.
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 study the curved geometry of noncommutative 4-tori $\mathbb{T}_θ^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of $\mathbb{T}_θ^4$. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local f…
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…
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…
A geometric Elliott invariant and noncommutative rigidity of mapping tori
本文利用非交换几何工具证明了极小拓扑动力系统映射环面的刚性，将保向叶状同伦等价提升为叶状C*-代数同构，并发展了基于拓扑与指标理论的几何Elliott不变量方法。
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。
We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz, are Leibniz quantum compact metric spaces and that they form a continuous family over the group of invertible matrices with entries in the commutant of the quantum tori in the regular representation, when this group is endowed with a natural length function.
We replaced the classical string theory notions of parameter space and world-time with noncommutative tori and consider maps between these spaces. The dynamics of mappings between different noncommutative tori were studied and a noncommutative generalization of the Polyakov action was derived. The quantum torus was studied in detail as well as *-homomorphisms between different quantum tori. A finite dimensional representation of the quantum torus was studied and the partition function and other path integrals were calculated. At the end we proved existence theorems for mappings between differe…
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…
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…
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.
We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas.
In the present investigation we are linking noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. Therefore we are in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this inves…
We continue the study of solitons over noncommutative tori from the perspective of time-frequency analysis and treat the case of a general topological charge. Solutions are associated with vector bundles of higher rank over noncommutative tori. We express these vector bundles in terms of vector-valued Gabor frames and apply the duality theory of Gabor analysis to show that Gaussians are solitons of general topological charge over noncommutative tori. An energy functional for projections over noncommutative tori is the basis for the self and anti-self duality equations of the solitons which tur…
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。
An analogue of Gross' logarithmic Sobolev inequality for a class of elements of noncommutative two tori is proved.
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。
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