Fredy Díaz García, Elmar Wagner
A Dirac operator is presented that will yield a 1+ summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes' conditions for noncommutative spin geometries are analyzed and it is argued that the failure of some requirements is mainly due to a wrong choice of a noncommutative spin bundle.
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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。
In this paper we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as direct limits of spectral triples on rotation algebras. In the final section we prove a noncommutative Wiener's lemma and show that our odd spec…
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。
Examples of noncommutative self-coverings are described, and spectral triples on the base space are extended to spectral triples on the inductive family of coverings, in such a way that the covering projections are locally isometric. Such triples are shown to converge, in a suitable sense, to a semifinite spectral triple on the direct limit of the tower of coverings, which we call noncommutative solenoidal space. Some of the self-coverings described here are given by the inclusion of the fixed point algebra in a C$^*$-algebra acted upon by a finite abelian group. In all the examples treated he…
We construct a 1+ summable regular even spectral triple for a noncommutative torus defined by a C*-subalgebra of the Toeplitz algebra.
It is well-known that any covering space of a Riemannian manifold has the natural structure of a Riemannian manifold. This article contains a noncommutative generalization of this fact. Since any Riemannian manifold with a Spin-structure defines a spectral triple, the spectral triple can be regarded as a noncommutative Spin-manifold. Similarly there is an algebraic construction which is a noncommutative generalization of topological covering. This article contains a construction of spectral triple on the "noncommutative covering space".
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 …
The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of algebras, and a spectral triple on a certain invariant subalgebra of the product algebra. These isomorphisms intertwine also the grading and real structure. This holds for all four inequivalent spin structures, which are explicitly constructed in terms of double coverings of the noncommutative torus (with arbitrary real parameter). These results are extended also t…
Comparison between two differential graded algebras in Noncommutative Geometry
本文比较了非交换几何中Connes和Fröhlich等人定义的两种微分格代数，发现Connes的dga在信息性上更具优势。
We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifo…
We discuss the Fubini formula in Alain Connes' noncommutative geometry. We present a sufficient condition on spectral triples for which a Fubini formula holds true. The condition is natural and related to heat semigroup asymptotics. We provide examples of spectral triples for which the Fubini formula fails.
We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.
It is well known that for any irrational rotation number $\a$, the noncommutative torus $\ba_\a$ must have representations $π$ such that the generated von Neumann algebra $π(\ba_\a)"$ is of type $\ty{III}$. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when $\a$ is a Liouville number satisfying a faster approximation prop…
Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.
A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
本研究针对谱截断谱三元组提出非交换积分的简单近似，推导得到非交换几何的Szegő极限公式，建立其与态密度的关联，提出紧谱三元组测地流遍历性的定义，揭示量子遍历性与Connes积分公式的紧密联系。
In previous papers, we constructed smooth (1,\infty)-summable semfinite spectral triples for graph algebras with a faithful trace, and (k,\infty)-summable semifinite spectral triples for k-graph algebras. In this paper we identify classes of graphs and k-graphs which satisfy a version of Connes' conditions for noncommutative manifolds.
It is proved that the (volume and orientation-preserving) quantum isometry group of a spectral triple obtained by deformation by some dual unitary 2-cocycle is isomorphic with a similar twist-deformation of the quantum isometry group of the original (undeformed) spectral triple. This result generalizes similar work by Bhowmick and Goswami for Rieffel-deformed spectral triples in [Comm. Math. Phys. 285 (2009), 421-444, arXiv:0707.2648].
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under co…
Let G be a finitely generated discrete group. The standard spectral triple on the group C*-algebra C*(G) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular the quantum isometry group of the C*-algebra of the free group on n-generators is computed and turns out to be a quantum group extension of the quantum permutation group A_{2n} of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and construction of the Laplacian for the standard spectral triple on C*…
Any oriented Riemannian manifold with a Spin-structure defines a spectral triple, so the spectral triple can be regarded as a noncommutative Spin-manifold. Otherwise for any unoriented Riemannian manifold there is the two-fold covering by oriented Riemannian manifold. Moreover there are noncommutative generalizations of finite-fold coverings. This circumstances yield a notion of unoriented spectral triple which is covered by oriented one.
A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A non-compact space is modelled in noncommutative geometry by a non-unital spectral triple. Our aim is to establish the Connes character formula for non-unital spectral triples. This is significantly more difficult than in the unital case and we achieve it with the use of recently developed double operator integration techniques. Previously, only partial extensions o…
Gelfand - Naĭmark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Similarly a spectral triple is a generalization of a Riemannian manifold. An (infinitely listed) covering of a Riemannian manifold has natural structure of Riemannian manifold. Here we will consider the noncommutative generalization of this result.
Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces $\R^{2N}$ endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes--Lott functional action, are given for these noncommutative hyperplanes.
Asymmetric noncommutative torus has vanishing Einstein tensor
研究通过部分共形缩放计算非交换 torus 上谱三元组的谱度量，发现其 torsion 和 Einstein 张量均消失。
The subject of this PhD thesis is noncommutative geometry - more specifically spectral triples - and how it can be generalized to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis will thus be dedicated to the transition from Riemannian to semi-Riemannian manifolds. This entails a study of Clifford algebras for indefinite vector spaces and Spin structures on semi-Riemannian manifolds. An important consequence of this is the introduction of Krein spaces, which will enable us to generalize spectral triples to indefinite spectral triples. I…
Parabolic noncommutative geometry
本文提出 tangled 谱三元组概念，用于研究抛物几何中的各向异性及非交换几何中的共形等价性，通过构造次椭圆谱三元组和交叉乘积谱三元组等方法实现。
We study the isospectral deformations of the Eguchi-Hanson spaces along a torus isometric action in the noncompact noncommutative geometry. We concentrate on locality, smoothness and summability conditions of the nonunital spectral triples, and relate them to the geometric conditions to be noncommutative spin manifolds.
After a review of the results in arXiv:1203.3184 [math-ph] about Pythagorean inequalities for products of spectral triples, I will present some new results and discuss classes of spectral triples and states for which equality holds.
The BV construction for finite spectral triples
本文研究BV形式如何自然嵌入由有限谱三元组诱导的规范理论非交换几何框架，证明BV和BRST复形与余代数的Hochschild复形重合，并详细展示了$U(n)$规范理论的构造。
To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when d is finite non-zero, there always exists a singular trace which is finite nonzero on |D|^-d, giving rise to a noncommutative integration on A. Such results are applied to fractals in R, using Connes' spectral triple, and to limit fractals in R^n, a class which generalises self-similar fractals, using a new spectral triple. The noncommutative dimension or measure c…
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.
Spectral triples over noncommutative principal $\T^n$-bundles are studied, extending recent results about the noncommutative geometry of principal U(1)-bundles. We relate the noncommutative geometry of the total space of the bundle with the geometry of the base space. Moreover, strong connections are used to build new Dirac operators. We discuss as a particular case the commutative case, the noncommutative tori and theta deformed manifolds.
Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sens…
We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier trace…
Collapse in Noncommutative Geometry and Spectral Continuity
本文研究非交换几何中谱三元组在坍缩下的谱连续性，提出沿垂直方向坍缩的方法，并借助谱邻近性获得收敛与连续性结果，涵盖乘积、主丛等例子。
We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we …
Two spectral triples are introduced for a class of fractals in R^n. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and Hausdorff-Besicovitch functionals considered in math.OA/0202108, are studied for the mentioned fractals endowed with these spectral triples, showing in many cases their correspondence with classical objects. In particular, for any limit fractal, the Hausdorff-Besicovitch functionals do not depend on the generalized limit procedure.
Noncommutative geometry on the Berkovich projective line
本文构建了与伯克维奇射影线相关的C*-代数和谱三元组，通过有限R树的极限构造谱三元组，并展示了如何利用这些代数获得不变测度。
We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.
For a locally compact quantum group $\G$ with tracial Haar weight $φ$, and a quantum measure $μ$ on $\G$, we study the space ${H}_μ^p$ of $μ$-harmonic operators in the non-commutative $L^p$-space ${L}^p(\G)$ associated to the Haar weight $φ$. The main result states that if $μ$ is non-degenerate, then ${H}_μ^p$ is trivial for all $1 \leq p < \infty$.
In attempts to combine non-commutative geometry and quantum gravity, Aastrup-Grimstrup-Nest construct a semi-finite spectral triple, modeling the space of G-connections for G=U(1) or SU(2). AGN show that the interaction between the algebra of holonomy loops and the Dirac-type operator D reproduces the Poisson structure of General Relativity in Ashtekar's loop variables. This article generalizes AGN's construction to any connected compact Lie group G. A construction of AGN's semi-finite spectral triple in terms of an inductive limit of spectral triples is formulated. The refined construction pe…
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…
Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two ex…
We present a new, general approach to gauge theory on principal $G$-spectral triples, where $G$ is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for $G$-$C^\ast$-algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded $KK^G$-cycle in the case of a principal $G$-action. Then, we introduce a notion of principal $G$-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded $KK^G$-theory with respect to such a cycle: up to a…
Let $T : Λ\to Λ$ be an expanding map on a Cantor set. For each suitably normalized Hölder continuous potential, we construct a spectral triple from which one may recover the associated Gibbs measure as a noncommutative measure.
We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent), i.e. as a generalised quantum limit. Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a v…
While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [x_i,x_j]=i theta_{ij}. Here we present new classes of (non-formal) deformed products associated to linear Lie algebras of the kind [x_i,x_j]=ic_{ij}^k x_k. For all possible three-dimensional cases, we define a new star product and discuss its properties. To complete the analysis of these novel noncommutative spaces, we introduce noncompact spectral triples, and the concept of star triple, a specialization of the spectral triple to deformations …
The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a Lévy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property …
We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudo-differential operators of order $-d$ and generalise A. Connes' trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order $-d$ do not have a `unique' trace; pseudo-differential operators can be non-measurable in Connes' sense. Other corollaries are given clarifying the role of Dixmier traces in …
We study some topological spaces that can be considered as hyperspaces associated to noncommutative spaces. More precisely, for a NC compact space associated to a unital C*-algebra, we consider the set of closed projections of the second dual of the C*-algebra as the hyperspace of closed subsets of the NC space. We endow this hyperspace with an analog of Vietoris topology. In the case that the NC space has a quantum metric space structure in the sense of Rieffel we study the analogs of Hausdorff and infimum distances on the hyperspace. We also formulate some interesting problems about distance…
We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximated by finite dimensional quantum compact metric spaces, and that they form a continuous family of quantum compact metric spaces over the space of multipliers of the solenoid, properly metrized.
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.
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。
Witten constructed a topological quantum field theory with the Chern-Simons action as Lagrangian. We define a Chern-Simons action for 3-dimensional spectral triples. We prove gauge invariance of the Chern-Simons action, and we prove that it concurs with the classical one in the case the spectral triple comes from a 3-dimensional spin manifold. In contrast to the classical Chern-Simons action, or a noncommutative generalization of it introduced by A. H. Chamseddine, A. Connes, and M. Marcolli by use of cyclic cohomology, the formula of our definition contains a linear term which shifts the crit…
We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.
We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) spectral triple. This computation is obtained as a straightforward consequence of a new proof of the local equivariant index theorem of Patodi, Donelly-Patodi and Gilkey. This proof is obtained by combining Getzler's rescaling with an equivariant version of Greiner's approach to …
Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence
本文在Latremoliere的propinquity框架下，从度量角度研究了谱三元组的等距群与双Lipschitz自同构集，证明了其紧致性，并给出了归纳极限下等距群作用在协变Gromov-Hausdorff propinquity意义下收敛的充要条件。
This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-dynamical $G$-system $\mathfrak{M}$ with compact subsystem $\mathfrak{N}$ is disjoint relative to $\mathfrak{N}$ from its maximal compact subsystem $\mathfrak{M}_{K}$ if and only if $\mathfrak{N}\cong\mathfrak{M}_{K}$. This generalizes recent work of Duvenhage, which established the result for $G$ abelian.
Fibrewise compactifications and generalised limits in commutative and noncommutative topology
本文在局部紧Hausdorff空间与$C^*$-代数设定下引入纤维紧致化，定义了保持局部紧性的受控极限，推广了经典逆极限并实现了乘子值$*$-同态的正向极限构造。
The bispectral problem is motivated by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admitting a second-order differential operator with a simple spectrum in its commutator. In this article, we discuss a noncommutative version of the bispectral problem, obtained by allowing all objects in the original formulation to be matrix-valued. Deep attention is given to bispectral algebras and their presentations as a tool to get information about bispectral triples.
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