The properties of Banach algebras affected by differential identities

TL;DR

This paper investigates how specific algebraic identities over open subsets influence the commutativity of Banach algebras, providing conditions and examples to justify the limitations of these results.

Contribution

It introduces new conditions under which certain identities imply the commutativity of Banach algebras, expanding understanding of algebraic structure implications.

Findings

Identifies conditions linking identities to commutativity

Provides examples illustrating limitations of the results

Establishes new criteria for algebraic commutativity

Abstract

Let $B$ be a Banach algebra. The interest of this article lies in the study of the commutativity of B if certain specific algebraic identities hold over a non-empty open subset of B. The limitations imposed in the hypothesis of our results are justified by some examples.