Mathematics

CLC number: O177.1

Nonlinear A*B+B*A Type Derivations on *-Algebras

¹ School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, Shaanxi, China
² Xi'an Modern Control Technology Institute, Xi'an 710065, Shaanxi, China

Received: 5 March 2023

Abstract

Let ${A}$ be a unital *-algebra with the unit $I$ and a nontrivial projection $P\in {A}$. Suppose that ${A}$ satisfies $X{A}P=\mathrm{0}$ implies $X=\mathrm{0}$ and $X{A}(I-P)=\mathrm{0}$ implies $X=\mathrm{0}.$ In this paper, we prove that $\varphi$ is a nonlinear $A^{\mathrm{*}}B+B^{\mathrm{*}}A$ type derivation on ${A}$ if and only if $\varphi$ is an additive *-derivation. This is then applied to prime *-algebra, von Neumann algebras with no central summands of type $I_{\mathrm{1}},$ factor von Neumann algebras and standard operator algebras.