Let 𝒜 be a Banach *-algebra. By 𝒮𝒜 we denote the set of all self-adjoint elements of 𝒜 and by 𝒪𝒜 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
$\overline {{\cal O}_{\cal A} } = {\cal S}_{\cal A }$
and {dn} is a sequence of uniformly bounded linear mappings satisfying
${\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 dn(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 dn = 0 for each n ≥ 1.