Unitary stable ranks and norm-one ranks

TL;DR

This paper investigates the relationships between various stable ranks in commutative $C^*$-algebras, introducing new concepts and providing conditions for norm-one reducibility, thereby advancing the understanding of algebraic and topological invariants.

Contribution

It establishes the equivalence of all-units rank, norm-one rank, and topological stable rank in commutative $C^*$-algebras and introduces the unitary $M$-stable rank for rings.

Findings

All-units rank and norm-one rank coincide with topological stable rank in commutative $C^*$-algebras.

Introduces the unitary $M$-stable rank and compares it with the Bass stable rank.

Provides a sufficient condition for norm-one reducibility in uniform algebras.

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.