Nonunital spectral triples and metric completeness in unbounded KK-theory
Bram Mesland, Adam Rennie
TL;DR
This paper develops a unified framework for understanding spectral metric spaces, connections on modules, and Kasparov products in unbounded KK-theory, by leveraging properties of approximate units and strengthening Kasparov's theorem.
Contribution
It introduces a novel approach to characterize completeness and construct unbounded Kasparov products, extending Kasparov's technical results.
Findings
Unified approach to spectral metric space completeness
Construction of unbounded Kasparov products
Strengthened Kasparov's technical theorem
Abstract
By considering the general properties of approximate units in differentiable algebras, we are able to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, and the lifting of Kasparov products to the unbounded category. In particular, by strengthening Kasparov's technical theorem, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product.
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