Topological invariance of the homological index

A. Carey, Jens Kaad
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引用次数: 9

Abstract

R. W. Carey and J. Pincus in [CaPi86] proposed and index theory for non-Fredholm bounded operators T on a separable Hilbert space H such that TT* - T*T is in the trace class. We showed in [CGK13] using Dirac-type operators acting on sections of bundles over R^{2n} that we could construct bounded operators T satisfying the more general condition that (1-TT*)^n - (1-T*T)^n is trace class. We proposed there a "homological" index for these Dirac-type operators given by Tr( (1-TT*)^n - (1-T*T)^n ). In this paper we show that the index introduced in [CGK13] represents the result of a pairing between a cyclic homology theory for the algebra generated by T and T* and its dual cohomology theory. This leads us to establish homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations.
同调指标的拓扑不变性
R. W. Carey和J. Pincus在[CaPi86]中提出了可分离Hilbert空间H上的非fredholm有界算子T的和指标理论,使得TT* - T*T在迹类中。我们在[CGK13]中表明,使用作用于R^{2n}上束的部分上的狄拉克型算子,我们可以构造出满足(1-TT*)^n - (1-T*T)^n是迹类的更一般条件的有界算子T。我们提出了这些狄拉克型算子的“同调”指标,由Tr((1-TT*)^n - (1-T*T)^n给出。本文证明了[CGK13]中引入的指标表示由T和T*生成的代数的一个循环同调理论与其对偶上同调理论之间的配对结果。这使我们建立了同伦指标的同伦不变性(在循环理论的意义上)。然后,我们能够以非常一般的方式定义某些无界算子的同调指标,并证明该指标在一类无界扰动下的不变性。
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