Tamaz Kandelaki
Kasparov $KK$-groups $KK(A,B)$ are represented as homotopy groups of the Pedersen-Weibel nonconnective algebraic $K$-theory spectrum of the additive category of Fredholm $(A,B)$-bimodules for $A$ and $B$, respectively, a separable and $σ$-unital trivially graded real or complex $C^*$-algebra acted upon by a fixed compact metrizable group.
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This paper has been withdrawn because it is a duplicate of [math/0609208].
The Chern character of $\vartheta$-summable $\mathcal{C}_q$-Fredholm modules over locally convex differential graded algebras
本文构造局部凸微分分次代数上 $\vartheta$-可和 $\mathcal{C}_q$-Fredholm 模的 Chern 特征标，统一奇偶情形并建立其与谱流的联系。
Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podleś spheres and from $SU_q(2)$.
We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in K-homology of graph C*-algebras. We prove that every K-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators; and we exhibit generating Fredholm modules for the K-homology of quantum lens spaces.
It is shown that Connes' character formula for unbounded, theta-summable Fredholm modules represents the abstract Chern-character in K-homology. As an application, the character of a particular Fredholm module over the reduced group C*-algebra of a discrete group is calculated, which leads to a partial solution of a problem posed by A.Connes.
We give a complete classification (up to smooth homotopy) of finitely summable Fredholm representations (Fredholm modules) over higher rank groups and lattices. Our results are a direct consequence of work of Bader, Furman, Gelander and Monod on generalisations of Kazhdan's property T for locally compact groups.
We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions' on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.
A combinatorial Fredholm module on self-similar sets built on $n$-cubes
本文在 Cantor 尘、Sierpinski 地毯等自相似集上构造了高维 Fredholm 模，推广了 Connes 的组合构造，并计算了其诱导算子的 Dixmier 迹。
Fredholm complexes of Hilbert C*-modules
本文研究以正则算子为微分的 Hilbert C*-模上链复形，刻画 Fredholm 性质，定义 K-理论值指标并证明其扰动稳定性及 Hodge 分解下的表达式。
We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the bounda…
Motivated by the search for new examples of "noncommutative manifolds", we study the noncommutative geometry of the group C*-algebras of various discrete groups. The examples we consier are the infinite dihedral group ${\bf Z} \times_σ {\bf Z}_2$ and the semidirect product ${\bf Z} \times_σ {\bf Z}$. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.
The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and com…
We prove that all K-homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K-d…
We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = SU_q(3)/T$ of $SU_q(3)$, based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold $ X_q $ satisfies Poincaré duality in equivariant $ KK $-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $SU_q(3)$.
We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd unbounded} $θ$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$. In the type $II_{\infty}$ case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzler's idea (suggested by I. M. Singer) of considering spectral flow (and eta i…
We introduce the notion of a {\vartheta}-summable Fredholm module over a locally convex dg algebra Ω and construct its Chern character as a cocycle on the entire cyclic complex of Ω, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack in the context of loop spaces. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type lo…
Analytic index theory and spectral flow in real Hilbert $C^*$-modules
本文在实Hilbert $C^*$-模中定义并研究Clifford反线性斜伴随Fredholm算子和自伴奇Fredholm算子的解析指标与谱流，利用Van Daele $K$-理论统一处理八种实$K$-群，并证明谱流的Robbin-Salamon定理。
We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $θ$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles …
To each algebra over the complex numbers we associate a sequence of abelian groups in a contravariant functorial way. In degree (m-1) we have the m-summable Fredholm modules over the algebra modulo stable m-summable perturbations. These new finitely summable K-homology groups pair with cyclic homology and algebraic K-theory. In the case of cyclic homology the pairing is induced by the Chern-Connes character. The pairing between algebraic K-theory and finitely summable K-homology is induced by the Connes-Karoubi multiplicative character.
In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an ordinary category of finitely generated projective modules on ordinary rings.
We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes-Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes-Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yield…
Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this talk I want to show how to deform the Fredholm family, in the sense of quantum groups. The family of Dirac type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transform covariantly under a (central extension of) the algebra.
We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C*-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules a…
Classical harmonic analysis viewed through the prism of noncommutative geometry
本文通过非交换几何框架，将经典调和分析与L^p空间分析结合，提出新的K-理论群和索引定理，揭示Hilbert变换和Riesz变换的作用。
A geometric representative for the fundamental class in KK-duality of Smale spaces
该研究针对非交换几何中Smale空间KK-对偶的基本类表示问题，通过推广Bowen马尔可夫划分构造受控利普希茨常数的动力单位分解，结合非周期点采样得到非交换惠特尼嵌入定理类似物，最终构造出表示基本类的θ-可和Fredholm模。
We show that all the projective modules over the coordinate ring of the real algebraic sphere of dimension 3 are free
The main purpose of this paper is computing higher algebraic $K$-theory of Koszul complexes over principal ideal domains. The second purpose of this paper is giving examples of comparison techniques on algebraic $K$-theory for Waldhausen categories without the factorization axiom.
In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of M.Levine-F.Morel claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism.
Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $π_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $π_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. T…
For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group theory has features of $K$-theory (e.g. determinant bundles, tame symbol cocycle for Kac-Moody extension), the $K$-theory torsor relates higher loop groups with higher $K$-theory. We study the classifying "index" map of this torsor in detail. We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the $K$-theory …
This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.
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