Spanier-Whitehead K-duality for $C^*$-algebras

TL;DR

This paper develops a noncommutative version of Spanier-Whitehead duality for certain $C^*$-algebras, extending classical duality concepts to the realm of operator algebras with finitely generated $K$-theory.

Contribution

It introduces Spanier-Whitehead $K$-duality for $C^*$-algebras, connecting it with Paschke duality and exploring its properties beyond initial assumptions.

Findings

Establishes Spanier-Whitehead $K$-duality for $C^*$-algebras with finitely generated $K$-theory.

Analyzes the relationship between Paschke duality and $K$-duality.

Explores the effects of relaxing initial assumptions on the duality theory.

Abstract

Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier-Whitehead $K$-duality, which is defined on the category of $C^*$-algebras whose $K$-theory is finitely generated and that satisfy the UCT with morphisms the $KK$-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier-Whitehead $K$-duality.