Certain properties of Jordan homomorphisms, n-Jordan homomorphisms and n-homomorphisms on rings and Banach algebras

Abstract

We investigate under what conditions n-Jordan homomorphisms between rings are n-homomorphism, or homomorphism; and under what conditions, n-Jordan homomorphisms are continuous. One of the main goals in this work is to show that every n-Jordan homomorphism f : A → B, from a unital ring A into a ring B with characteristic greater than n, is a multiple of a Jordan homomorphism and hence, it is an n-homomorphism if every Jordan homomorphism from A into B is a homomorphism. In particular, if B is an integral domain whose characteristic is greater than n, then every n-Jordan homomorphism f : A → B is an n-homomorphism. Along with some other results, we show that if A and B are unital rings such that the characteristic of B is greater than n, then every unital n-Jordan homomorphism f : A → B is a Jordan homomorphism and hence, it is an m-Jordan homomorphism for any positive integer m ≥ 2. We also investigate the automatic continuity of n-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the n-Jordan homomorphism has dense range.