Unitary stable ranks and norm-one ranks

Abstract: In the context of commutative $C^*$-algebras we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary $M$-stable rank for an arbitrary commutative unital ring and compare it with the Bass stable rank. In case of uniform algebras, a sufficient condition for norm-one reducibility is given.