Universal cycles and homological invariants of locally convex algebras

TL;DR

This paper develops a framework using locally convex Kasparov modules to induce isomorphisms in various functors on locally convex algebras, leading to new results like Bott periodicity and Thom isomorphism for Schwartz function algebras.

Contribution

It introduces a novel approach to relate algebraic K-theory, Kasparov modules, and functorial isomorphisms in the setting of locally convex algebras, extending classical results.

Findings

Proves Bott periodicity for Schwartz function algebras

Establishes Thom isomorphism in the locally convex setting

Demonstrates functorial isomorphisms via Kasparov modules

Abstract

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes-Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions.