$K$-theory of rings of continuous functions

TL;DR

This paper investigates the algebraic K-theory of rings of continuous functions on compact spaces, establishing negative K-theory computations, K-regularity, and confirming prior claims in real cases, with methods applicable to nonarchimedean and noncommutative cases.

Contribution

It provides new computations and proofs for the algebraic K-theory of continuous function rings, including previously unconfirmed claims in the real case, using algebraic methods applicable to broader contexts.

Findings

Computed negative K-theory for these rings.

Established K-regularity of the rings.

Confirmed two unproven claims in the real case.

Abstract

We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case reproves the results of Rosenberg, Friedlander--Walker, and Corti\~nas--Thom. Our consideration in the real case proves two previously unconfirmed claims made by Rosenberg in 1990. The algebraic nature of our methods enables us to deal with the nonarchimedean and noncommutative cases analogously.