On the topological stable rank of non-selfadjoint operator algebras

TL;DR

This paper demonstrates that for certain classes of non-selfadjoint operator algebras, the right and left topological stable ranks can differ or be infinite, providing counterexamples to previously assumed equivalences.

Contribution

It offers the first known negative answer to whether the right and left topological stable ranks of a Banach algebra must always coincide, using examples from nest algebras.

Findings

Counterexamples in nest algebras show differing stable ranks

Many nest algebras have infinite stable ranks on both sides

Results extend to non-commutative disc and free semigroup algebras

Abstract

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.