(Semi)topological $K$-theory via solidification

TL;DR

This paper shows that solidification in condensed mathematics unifies and recovers known constructions of semitopological and topological K-theory for real algebras, providing a new perspective.

Contribution

It demonstrates that solidification of algebraic K-theory encompasses both semitopological and topological K-theories, linking them through condensed mathematics.

Findings

Solidification recovers semitopological K-theory for real algebras.

Solidification recovers topological K-theory for real Banach algebras.

Unifies different K-theory constructions within condensed mathematics.

Abstract

Clausen--Scholze introduced the notion of solid spectrum in their condensed mathematics program. We demonstrate that the solidification of algebraic $K$-theory recovers two known constructions: the semitopological $K$-theory of a real (associative) algebra and the topological (aka operator) $K$-theory of a real Banach algebra.