Linear mappings satisfying some recursive sequences

Abstract

Let A be a unital, complex normed ∗-algebra with the identity element e such that the set of all algebraic elements of A is norm dense in the set of all self-adjoint elements of A and let {Dn} ∞n=0 and {∆n} ∞n=0 be sequences of continuous linear mappings on A satisfying    Dn+1(p) = ∑n k=0 Dn−k(p)Dk(p), ∆n+1(p) = ∑n k=0 ∆n−k(p)Dk(p), for all projections p of A and all nonnegative integers n. Moreover, suppose that D0(p) = D0(p) 2 holds for all projections p of A. Then ∆n = Cn 2 ( RD0(e)∆0 + L∆0(e)D0 ) for all n ∈ N, where Cn denotes the nth Catalan number and RD0(e)(a) = aD0(e) and L∆0(e)(a) = ∆0(e)a for all a ∈ A. Using this result, we present a characterization of left τ -centralizers satisfying a certain recursive relation. In addition, a characterization of generalized higher derivations is presented. Moreover, we show that higher derivations, prime higher derivations, left higher derivations, and σ -derivations are zero under certain conditions.