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

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 $X_{\mathrm{\infty }}$ constructed by Tarbard (Ph.D. Thesis, University of Oxford, 2013), the algebra of operators $B(X_{\mathrm{\infty }})$ is Dedekind-finite but does not have stable rank one. While this sheds some light on the Banach space structure of $X_{\mathrm{\infty }}$ 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.