A Banach space whose algebra of operators is Dedekind-finite but it does not have stable rank one

TL;DR

This paper constructs a Banach space whose algebra of operators is Dedekind-finite but lacks stable rank one, highlighting nuanced differences in operator algebra properties on specific Banach spaces.

Contribution

It demonstrates that the algebra of operators on a particular indecomposable Banach space can be Dedekind-finite without having stable rank one, providing a counterexample to potential assumptions.

Findings

The algebra of operators on $X_{ ext{infty}}$ is Dedekind-finite.

This algebra does not have stable rank one.

The property does not hold for all indecomposable Banach spaces.

Abstract

In this note we examine the connection between the stable rank one and Dedekind-finite property of the algebra of operators on a Banach space $X$. We show that for the indecomposable but not hereditarily indecomposable Banach space $X_{\infty}$ constructed by Tarbard (Ph.D. Thesis, University of Oxford, 2013), the algebra of operators $B(X_{\infty})$ is Dedekind-finite but does not have stable rank one. While this sheds some light on the Banach space structure of $X_{\infty}$ itself, we observe that the indecomposable but not hereditarily indecomposable Banach space constructed by Gowers and Maurey (Math. Ann., 1997) does not possess this property.