Projective Freeness and Stable Rank of Algebras of Complex-valued BV Functions

TL;DR

This paper studies the algebraic structure of Banach algebras of complex-valued bounded variation functions, establishing conditions for projective freeness and stable rank, based on cohomology and polynomial convexity.

Contribution

It introduces new results linking cohomology vanishing to algebraic properties of function algebras, specifically for BV functions.

Findings

Banach algebras of BV functions have Bass stable rank one.

Such algebras are projective free if they lack nontrivial idempotents.

Derived from a new cohomology vanishing result related to polynomial convexity.

Abstract

The paper investigates the algebraic properties of Banach algebras of complex-valued functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank one and are projective free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second \v{C}ech cohomology group of the polynomially convex hull of a continuum of a finite linear measure.