Banach Algebras with Bounded Groups of Generators, and the Schur Property

Recall that a Banach space $X$ is said to have the Schur property if any weakly compact set in $X$ is strongly compact. In this note we consider a Banach algebra $A$ that has a bounded group of generators. Along with other results, it is proved that if $A^*$ has the Schur property, then the Gelfand space of the algebra $A$ is a scattered set and, moreover, $A^*$ has the Radon–Nikodym property.