Semi-simplicity of some semi-prime Banach algebras.
In this survey, we summarise some of the recent progress on the structure of spectral isometries between C*-algebras.
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.