Noncommutative correspondence categories, simplicial sets and pro $C^*$-algebras

TL;DR

This paper explores the relationships between $KK$-theory, DG categories, and pro $C^*$-algebras, establishing functorial constructions and proposing a noncommutative proper homotopy theory.

Contribution

It introduces a correspondence between $KK$-equivalence and $ extbf{K}_*$-equivalence of DG categories and constructs a functorial pro $C^*$-algebra from simplicial sets.

Findings

$KK$-equivalence induces $ extbf{K}_*$-equivalence between DG categories

Pro $C^*$-algebra construction is functorial and respects proper homotopy

Proposes a noncommutative proper homotopy theory

Abstract

We show that a $KK$-equivalence between two unital $C^*$-algebras produces a correspondence between their DG categories of finitely generated projective modules which is a $\mathbf{K}_*$-equivalence, where $\mathbf{K}_*$ is Waldhausen's $K$-theory. We discuss some connections with strong deformations of $C^*$-algebras and homological dualities. Motivated by a construction of Cuntz we associate a pro $C^*$-algebra to any simplicial set. We show that this construction is functorial with respect to proper maps of simplicial sets, that we define, and also respects proper homotopy equivalences. We propose to develop a noncommutative proper homotopy theory in the context of topological algebras.