Approximate Riesz Algebra-Valued Derivations

Abstract

Let $F$ be a Riesz algebra with an extended norm $||·|{|}_u$ such that $(F,||·|{|}_u)$ is complete. Also, let $||·|{|}_v$ be another extended norm in $F$ weaker than $||·|{|}_u$ such that whenever (a) $x_n\to x$ and $x_n·y\to z$ in $||·|{|}_v$, then $z=x·y$; (b) $y_n\to y$ and $x·y_n\to z$ in $||·|{|}_v$, then $z=x·y$. Let $\epsilon$ and $\delta >$ be two nonnegative real numbers. Assume that a map $f:F\to F$ satisfies $||f(x+y)-f(x)-f(y)|{|}_u\le \epsilon$ and $||f(x·y)-x·f(y)-f(x)·y|{|}_v\le \delta$ for all $x,y\in F$. In this paper, we prove that there exists a unique derivation $d:F\to F$ such that $||f(x)-d(x)|{|}_u\le \epsilon$, ($x\in F$). Moreover, $x·(f(y)-d(y))=0$ for all $x,y\in F$.