Approximate Riesz Algebra-Valued Derivations

Abstract EN

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) xn→x and xn·y→z in ||·||v, then z=x·y; (b) yn→y and x·yn→z in ||·||v, then z=x·y.

Let ε and δ> be two nonnegative real numbers.

Assume that a map f:F→F satisfies ||f(x+y)-f(x)-f(y)||u≤ε and ||f(x·y)-x·f(y)-f(x)·y||v≤δ for all x,y∈F.

In this paper, we prove that there exists a unique derivation d:F→F such that ||f(x)-d(x)||u≤ε, (x∈F).

Moreover, x·(f(y)-d(y))=0 for all x,y∈F.