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

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.

Version 2. References added, Koszmider space example added. To appear in Proc. 24th International Conference on Banach algebras and Applications. 9 pp

References in corpus (2)

• Operators on Banach Spaces of Bourgain-Delbaen Type
• The multiplier algebra of the noncommutative Schwartz space

Cited by in corpus (1)

• Ring-theoretic (in)finiteness in reduced products of Banach algebras