Commutativity Criteria for Prime Banach Algebras via Local Differential Identities

This paper establishes new commutativity criteria for real or complex prime Banach algebras [Formula: see text] admitting a non-injective continuous derivation [Formula: see text]. By assuming that one of the local differential identities [Formula: see text], [Formula: see text], or [Formula: see text] holds for all [Formula: see text] in a nonempty open subset [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text] and [Formula: see text], we prove that [Formula: see text] must be commutative. Using Baire’s category theorem combined with polynomial identity methods and ring theory, we extend these local centrality conditions to the entire algebra. Finally, explicit matrix counterexamples demonstrate that both the openness of the subsets and the choice of the field [Formula: see text] or [Formula: see text] are essential for the validity of these results.