Algebraic $K$-theory Spectra and Factorisations of Analytic Assembly Maps

TL;DR

This paper develops a unified framework for algebraic and analytic K-theory spectra associated with topological ringoids, enabling factorization of assembly maps relevant to major conjectures like Novikov and Baum-Connes.

Contribution

It introduces connective K-theory spectra for topological ringoids and demonstrates how analytic assembly maps can be factorized using this framework.

Findings

Unified algebraic and analytic K-theory spectra for topological ringoids.

Factorization of assembly maps in Novikov and Baum-Connes conjectures.

Application of existing machinery to new spectral constructions.

Abstract

In this article we use existing machinery to define connective $K$-theory spectra associated to topological ringoids. Algebraic $K$-theory of discrete ringoids, and the analytic $K$-theory of Banach categories are obtained as special cases. As an application, we show how the analytic assembly maps featuring in the Novikov and Baum-Connes conjectures can be factorised into composites of assembly maps resembling those appearing in algebraic $K$-theory and maps coming from completions of certain topological ringoids into Banach categories. These factorisations are proved by using existing characterisations of assembly maps along with our unified picture of algebraic and analytic $K$-theory.