Dense subsets on Banach *-algebras with linear derivations

Let A be a Banach ∗-algebra over C. In this manuscript, we study the behaviour of linear derivations with regular involution which satisfy certain differential identitities. In fact, we prove that there is no positive integer n such that the set of a ∈ A for which (a∆)n((a∗)∆)n ± ((a∗)∆)n(a∆)n ∈ Z(A ) or there exists a central idempotent e ∈ Q such that ∆ = 0 on eQ and (1 − e)Q satisfies s4, the standard identity in four variables. Mathematics Subject Classification: 16W25, 46J45