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

摘要

Let B be a semiprime commutative unital Banach algebra with connected character space Phi(B). For each x is an element of Phi(B,) let pi(B)(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 pi(B)(x) at each x is an element of Phi(B) 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. (C) 2016 The Authors. Published by Elsevier Inc.