A Banach space whose algebra of operators is Dedekind-finite but it does not have stable rank one · arXivDesk
1807.10578Jul 27, 2018Version 2. References added, Koszmider space example added. To appear in Proc. 24th International Conference on Banach algebras and Applications. 9 pp
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∞ constructed by Tarbard (Ph.D. Thesis, University of Oxford, 2013), the algebra of operators B(X∞)
Nearby in the stack
is Dedekind-finite but does not have stable rank one. While this sheds some light on the Banach space structure of
X∞
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.