  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.
