Differential embeddings into algebras of topological stable rank 1

TL;DR

This paper characterizes when multiplication is open in certain Banach *-algebras and function algebras, linking algebraic properties to topological invariants like stable rank and covering dimension.

Contribution

It introduces a class of smooth Banach *-algebras with differential subalgebra properties and characterizes openness of multiplication in function algebras via topological dimension.

Findings

Group algebras of unbounded exponent lack uniformly open convolution

Openness of multiplication in function algebras relates to covering dimension

Identifies differential subalgebras with stable rank 1

Abstract

We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterise in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.