Additive Maps and a Criterion for Integrability with Quasi-Compact Endomorphisms

Abstract

We show that Φ preserves zero products in both directions if and only if Φ is either an isomorphism or a conjugate isomorphism .We arrive at the same conclusion for an arbitraryunital, complex Banach algebra, by imposing an extra surjectivity condition on the map. Let 𝐺 be a reductive group and 𝜃 an involution on 𝐺, both defined over a p-adicfild. We provide a criterion for 𝐺𝜃-integrability of matrix coefficients of representations of 𝐺 in terms of their exponents along 𝜃-stable parabolic subgroups.Let 𝐵 be a semiprime commutative unitalBanach algebra with connected character space Φ𝐵.For each 𝑥 𝜖 Φ𝐵, let π𝐵(𝑥) be the collection of all closed primary ideals contained in the maximal ideal 𝑀(𝑥)= 𝑥−1(0). The purpose is to illustrate how knowledge of the collection π𝐵(𝑥) at each𝑥 𝜖 Φ𝐵can be used in describing the outer spectrum of a quasi-compact unital endomorphism of 𝐵.