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

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.