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Let R be a 2-torsion free σ-prime ring, U a nonzero σ-square closed Lie ideal of R and d a derivation of R which commutes with σ. If d acts as a homomorphism or an anti-homomorphism on U, then either d = 0 or U ⊆ Z(R).Mathematics Subject Classification: 16W10, 16W25, 16U80.

In this note we investigate left multipliers satisfying certain algebraic identities on Lie ideals of rings with involution and discuss related results. Moreover we provide examples to show that the assumed restrictions cannot be relaxed.

In this paper, we investigate the commutativity of a Banach algebra [Formula: see text] provided with a continuous derivation satisfying algebraic identities involving nonvoid open subsets of [Formula: see text] Furthermore, we provide examples to show that various restrictions in the hypothesis of our theorems are not superfluous.