Banach spaces whose algebras of operators are Dedekind-finite but they do not have stable rank one

摘要

In this note we examine the connection between the stable rank oneand Dedekind-finite property of the algebra of operators on a Banach space X.We show that for the complex indecomposable but not hereditarily indecomposableBanach space X_∞ constructed by Tarbard (Ph.D. Thesis, University of Oxford,2013), the algebra of operators B(X_∞) is Dedekind-finite but does not have stablerank one. While this sheds some light on the Banach space structure of X_∞ itself,we observe that the indecomposable but not hereditarily indecomposable Banachspace constructed by Gowers and Maurey (Math. Ann., 1997) does not possess thisproperty. We also show that if K is the connected "Koszmider" space constructedby Plebanek in ZFC (Topology and its Applications, 2004), then B(C(K, R)) isDedekind-finite but does not have stable rank one.