Stability of commuting maps and Lie maps
Let $A$ be an ultraprime Banach algebra. We prove that each approximately commuting continuous linear (or quadratic) map on $A$ is near an actual commuting continuous linear (resp. quadratic) map on $A$. Furthermore, we use this analysis to study how close are approximate Lie isomorphisms and approximate Lie derivations to actual Lie isomorphisms and Lie derivations, respectively.