Approximate Riesz algebra-valued derivations
Summary: Let \(F\) be a Riesz algebra with an extended norm \(||\cdot||_u\) such that \((F, ||\cdot||_u)\) is complete. Also, let \(||\cdot||_v\) be another extended norm in \(F\) weaker than \(||\cdot||_u\) such that, whenever (a) \(x_n \rightarrow x\) and \(x_n \cdot y \rightarrow z\) in \(||\cdot||_v\), then \(z = x \cdot y\); (b) \(y_n \rightarrow y\) and \(x \cdot y_n \rightarrow z\) in \(||\cdot||_v\), then \(z = x \cdot y\). Let \(\varepsilon\) and \(\delta\) be two nonnegative real numbers. Assume that a map \(f : F \rightarrow F\) satisfies \(||f(x + y) - f(x) - f(y)||_u \leq \varepsilon\) and \(||f(x \cdot y) - x \cdot f(y) - f(x) \cdot y||_v \leq \delta\) for all \(x, y \in F\). In this paper, we prove that there exists a unique derivation \(d : F \rightarrow F\) such that \(||f(x) - d(x)||_u \leq \varepsilon\) \((x \in F)\). Moreover, \(x \cdot (f(y) - d(y)) = 0\) for all \(x, y \in F\).
scientific article; zbMATH DE number 6116353