Michael Puschnigg
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.
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We prove that odd unbounded p-summable Fredholm modules are also bounded p-summable Fredholm modules (this is the odd counterpart of a result of A. Connes for the case of even Fredholm modules).
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…
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 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…
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.
Finitely Summable $K$-homology, the Index Pairing, and Cantor Minimal Systems
本文研究 Cantor 极小系统交叉积 $C^*$-代数的指标配对，利用 Putnam 轨道破缺 AF-子代数证明可通过 Connes 迹公式计算任意配对，并证明里程计代数具有一致有限可和 $K$-同调。
Let (A,H,F) be a p-summable Fredholm module where the algebra A= C Γis generated by a discrete group of unitaries in L(H) which is of polynomial growth r. Then we construct a spectral triple (A,H,D) with F= sign D which is q-summable for each q > p+r+1. In case (A,H,F) is (p,\infty)-summable we obtain (q,\infty)-summability of (A,H,D) for each q > p+r+1.
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.
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 …
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…
Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*-algebras associated to hyperbolic groups - the C*-crossed product for the boundary action, and the reduced group C*-algebra - have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit fini…
The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.
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 study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is le…
In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods…
We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C…
A Riesz-Fredholm type theorem on certain Hilbert C*-modules
本文研究满足性质 $\mathbb{[H]}$ 的 Hilbert C*-模上紧模算子 $C$ 与 $I-C$ 的 Riesz-Fredholm 理论，证明存在唯一幂次使 $L^r$ 为 EP 算子，并给出核与值域的直和分解及方程可解性框架。
In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Λ$, via the infinite path space $Λ^\infty$ of $Λ$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Λ^\infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $Λ$. We then study the zeta…
Super $K$-theory and group completion
该研究构建实超巴拿赫代数的谱层面分次K-理论，采用范畴与同伦论方法，定义相关谱并证明其性质，建立Bott周期性与Morita等价的联系，还给出经典K群的谱细化及非分次拓扑环K-理论的泛性质刻画。
Elliptic Boundary Value Problems and Partial Group Actions
本文研究具有边界的紧致光滑流形上可群的部分作用，通过球面爆破构造流形Y及其部分作用，定义C*-代数并给出椭圆算子Fredholm性的判据，以及椭圆元素在稳定同伦下的分类。
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