The fine structure of the Kasparov groups I: continuity of the KK-pairing

A Note on Kasparov Product and Duality

Using Paschke-Higson duality, we can get a natural index pairing $K_{i}(A) \times K_{i+1}(D_Φ) \to \boldsymbol{Z} \quad (i=0,1) (\mbox{mod}2)$, where $A$ is a separable $C\sp*$-algebra, and $Φ$ is a representation of $A$ on a separable infinite dimensional Hilbert space $H$. It is proved that this is a special case of the Kasparov Product. As a step, we show a proof of Bott-periodicity for KK-theory asserting that $\mathbb{C}_1$ and $S$ are $KK$-equivalent using the odd index pairing.

通过渐近态射和准投影的奇谱局部化

The odd spectral localiser via asymptotic morphisms and quasi-projections

本文通过渐近态射和准投影研究奇K理论类与奇无界Kasparov模的索引配对，提出奇谱局部化的实例。

Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases

In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable $C^*$-algebra by a twisted $\mathbb{R}^d$-action. The spectral triple allows us to employ the non-unital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In addition, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener-Ho…

关于权重、迹与K理论

On weights, traces and K-theory

本文证明了C*-代数的K00群与稠密定义的迹的配对可扩展至权重，且迹的配对可无需半连续性假设地扩展到K0群。

KK-理论中的谱局部化子

Spectral localizers in KK-theory

本文研究偶KK-理论中由Kasparov积导出的偶K-群指标同态，利用无界Kasparov模块和谱局部化子给出了不依赖完整谱的显式公式，并推广了Loring和Schulz-Baldes的指标配对工作。