Quantitative $K$-theory for Banach algebras

TL;DR

This paper extends quantitative $K$-theory to algebras of bounded operators on subquotients of $L_p$ spaces, providing a new framework and a controlled Mayer-Vietoris sequence for these algebras.

Contribution

It develops a quantitative $K$-theory framework for operator algebras on subquotients of $L_p$ spaces and establishes a controlled Mayer-Vietoris sequence.

Findings

Framework for quantitative $K$-theory on subquotients of $L_p$ spaces

Existence of a controlled Mayer-Vietoris sequence in this setting

Extension of previous work on $C^*$-algebras to a broader class of Banach algebras

Abstract

Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Herv\'e Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework.