Characterizations of Jordan derivations and Jordan homomorphisms

Abstract

Let 𝒜 be a unital Banach algebra and ℳ be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ℳ satisfying δ(ST) = δ(S)T +Sδ(T) for any S, T ∈ 𝒜 with ST = W, where W is a left or right separating point of ℳ, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ℬ which satisfies h(S)h(T) = h(ST) for any S, T ∈ 𝒜 with ST = W is a Jordan homomorphism if h(W) is a separating point of ℬ.

Keywords:

• CSL algebra
• derivation
• Jordan derivation
• Jordan homomorphism

AMS Subject Classifications:

• 47B47
• 47L35

Acknowledgement

This article is supported by the National Science Foundation of China.