Homotopical stable ranks for Banach algebras

TL;DR

This paper explores the properties and computations of homotopical stable ranks in Banach algebras, revealing their strong similarities and differences with dimensional stable ranks.

Contribution

It provides a detailed study of the two homotopical stable ranks, highlighting their properties, examples, and the contrast with dimensional invariants.

Findings

Homotopical stable ranks are homotopy invariants for Banach algebras.

There is a strong affinity between the two homotopical stable ranks.

Homotopical stable ranks differ markedly from dimensional stable ranks.

Abstract

The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.