Stable rank for inclusions of Banach algebras

TL;DR

This paper derives a formula for the stable rank of inclusions of unital Banach algebras with finite Watatani index and applies it to compute the stable rank of certain group actions on Disk algebras.

Contribution

It introduces a formula for the stable rank in the context of Banach algebra inclusions with finite Watatani index and applies it to specific algebraic structures.

Findings

Stable rank of inclusions with finite Watatani index is explicitly characterized.

The stable rank of $ ext{l}^1$-algebras of Disk algebras under finite group actions is 2.

Provides a method to compute stable ranks in Banach algebra inclusions.

Abstract

We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.