Quasi-compact endomorphisms and primary ideals in commutative unital Banach algebras

Let B be a semiprime commutative unital Banach algebra with connected character space ΦB. For each x∈ΦB, let πB(x) be the collection of all closed primary ideals contained in the maximal ideal M(x)=x−1(0). The purpose of this paper is to illustrate how knowledge of the collection πB(x) at each x∈ΦB can be used in describing the outer spectrum of a quasi-compact unital endomorphism of B. Among other things, our results lead to the observation that when B is strongly regular, every Riesz endomorphism of B is quasi-nilpotent on an invariant maximal ideal. Some of the implications of our work for various other types of function algebra are explored at the end of the paper.