Jordan τ-Derivations of Locally Matrix Rings

Abstract

Let R be a prime, locally matrix ring of characteristic not 2 and let Q _ms (R) be the maximal symmetric ring of quotients of R. Suppose that ${\delta}\colon R\to Q_{ms}(R)$ is a Jordan τ-derivation, where τ is an anti-automorphism of R. Then there exists a ∈ Q _ms (R) such that δ(x) = xa − aτ(x) for all x ∈ R. Let X be a Banach space over the field ${\mathbb F}$ of real or complex numbers and let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on X. We prove that $Q_{ms}({\mathcal B}(X))={\mathcal B}(X)$, which provides the viewpoint of ring theory for some results concerning derivations on the algebra ${\mathcal B}(X)$. In particular, all Jordan τ-derivations of ${\mathcal B}(X)$ are inner if $\text{dim}_{\mathbb F}X>1$.