Some conditions under which derivations are zero on Banach *-algebras

Let 𝒜 be a Banach *-algebra. By 𝒮_𝒜 we denote the set of all self-adjoint elements of 𝒜 and by 𝒪_𝒜 we denote the set of those elements in 𝒜 which can be represented as finite real-linear combinations of mutually orthogonal projections. The main purpose of this paper is to prove the following result:

Suppose that 𝒪𝒜¯=𝒮𝒜$\overline {{\cal O}_{\cal A} } = {\cal S}_{\cal A }$ and {d_n} is a sequence of uniformly bounded linear mappings satisfying dn(p)=∑k=0ndn−k(p)dk(p)${\rm{d}}_{\rm{n}} ({\rm{p}}) = \sum\nolimits_{{\rm{k}} = 0}^{\rm{n}} {{\rm{d}}_{{\rm{n}} - {\rm{k}}} ({\rm{p}}){\rm{d}}_{\rm{k}} ({\rm{p}})} $ , where p is an arbitrary projection in 𝒜. Then d_n(𝒜) ⊆ ∪_ϕ∈Φ𝒜 ker ϕ for each n ≥ 1. In particular, if 𝒜 is semi-prime and further, dim(∪_ϕ∈Φ𝒜 ker ϕ) ≤ 1, then d_n = 0 for each n ≥ 1.