kk-Theory for Banach Algebras I: The Non-Equivariant Case

TL;DR

This paper introduces kk$^{ ext{ban}}$, a new bivariant K-theory for Banach algebras, establishing its foundational properties and universal construction via triangulated categories.

Contribution

It defines kk$^{ ext{ban}}$ through a universal property, ensuring its existence and connecting it with Lafforgue's KK$^{ ext{ban}}$ theory.

Findings

kk$^{ ext{ban}}$ has homological properties, a product, and Morita invariance.

The theory is constructed via a universal property using triangulated categories.

A natural transformation from Lafforgue's KK$^{ ext{ban}}$ to kk$^{ ext{ban}}$ is established.

Abstract

kk$^{\text{ban}}$ is a bivariant K-theory for Banach algebras that has reasonable homological properties, a product and is Morita invariant in a very general sense. We define it here by a universal property and ensure its existence in a rather abstract manner using triangulated categories. The definition ensures that there is a natural transformation from Lafforgue's theory KK$^{\text{ban}}$ into it so that one can take products of elements in KK$^{\text{ban}}$ that lie in kk$^{\text{ban}}$.