On the topology of the group of invertible elements

TL;DR

This paper surveys the homotopy theory of the invertible elements in Banach algebras, focusing on unstable K-Theory of real and complex C*-algebras, highlighting topological properties and recent developments.

Contribution

It provides a comprehensive overview of the topology of invertible elements in Banach algebras with emphasis on unstable K-Theory for C*-algebras, integrating recent research insights.

Findings

Detailed analysis of the homotopy groups of invertible elements

Connections between topology and K-Theory in Banach algebras

Identification of key topological invariants in C*-algebras

Abstract

We give a survey on the homotopy theory of the regular group of Banach algebras with emphasis on the unstable K-Theory of real and complex C*-algebras