The $K$-theory spectrum of the reduced group $C^\ast$-algebra is a functor

TL;DR

This paper constructs controlled $C^ ext{-}*$-categories to demonstrate that the $K$-theory spectrum of a reduced group $C^ ext{-}*$-algebra is a functor, establishing a functorial relationship in certain cases.

Contribution

It introduces controlled $C^ ext{-}*$-categories and proves the functoriality of the $K$-theory spectrum of reduced group $C^ ext{-}*$-algebras in specific scenarios.

Findings

Reduced group $C^ ext{-}*$-algebras have $K$-theory spectra that are functorial.

Constructed $C^ ext{-}*$-categories analogous to controlled algebraic $K$-theory categories.

Showed equivalence of $K$-theory spectra between the algebra and the associated controlled $C^ ext{-}*$-category.

Abstract

We construct $C^\ast$-categories that are anologues of the categories used in controlled algebraic $K$-theory. We then show that the reduced $C^\ast$-algebra of a finitely presented group and an associated controlled $C^\ast$-category have equivalent $K$-theory spectra, and that, at least in certain special cases, the associated $C^\ast$-category depends functorially on the group. Thus in these cases the $K$-theory spectrum of the reduced group $C^\ast$-algebra is a functor.