Exploring Commutativity and Endomorphic Structures in Banach Algebras: An Algebraic and Topological Analysis

This paper examines how endomorphisms influence the commutativity property within Banach algebras. We combine topological methods from functional analysis with algebraic techniques to derive conditions under which a prime Banach algebra becomes commutative. Central to our study is the role of a continuous surjective endomorphism that satisfies particular algebraic relations involving powers and commutators or Jordan-type products on non-empty open subsets of the algebra. Leveraging Baire’s category theorem, continuity arguments, and polynomial coefficient extraction, we establish global commutativity from these local conditions. Additionally, we present concrete counterexamples to demonstrate that key assumptions-such as the openness of the subsets, continuity and surjectivity of the endomorphism, the primeness of the algebra, and the nature of the underlying scalar field-are indispensable. Our findings build upon earlier work involving derivations and power commutators, offering fresh perspectives on the interplay between topology and algebra in forcing structural rigidity in noncommutative settings.