Triple derivable maps on prime algebras with involution

Abstract

Let A be a unital prime *-algebra with characteristic not 2 and with a projection $e\ne 0,1$. Suppose that $d:A\to Q_s(A)$ is a map satisfying $$\begin{array}{c}d\left(\frac{x{y}^{*}z+z{y}^{*}x}{2}\right)\\ =\frac{d(x)y^{*}z+xd{(y)}^{*}z+x{y}^{*}d(z)+d(z)y^{*}x+zd{(y)}^{*}x+z{y}^{*}d(x)}{2}\end{array}$$ for all $x,y,z\in A$, where $Q_s(A)$ denotes the symmetric Martindale algebra of quotients of A. It is shown that d is an additive triple derivation. Moreover, there exist $a\in Q_s(A)$ with $a^{*}=-a$ and an additive *-derivation $\delta :A\to Q_s(A)$ such that $d(x)=ax+\delta (x)$ for all $x\in A$. The analogous results for prime locally matrix *-algebras, prime *-algebras with nonzero socle, factor von Neumann algebras and standard operator *-algebras on Hilbert spaces are also described.

KEYWORDS:

• Functional identity
• involution
• prime algebra
• standard operator algebra
• triple derivation

2020 Mathematics Subject Classification:

• 16N60
• 16W10
• 16W25
• 16R60
• 47B47
• 46L57

Acknowledgments

The authors would like to thank the referee for the very thorough reading of the paper and valuable comments.