Stability of the van der Waerden theorem on the continuity of homomorphisms of compact semisimple Lie groups (Q884151)

Let \(G\) be a compact semisimple Lie group and let \(U(N)\) be the group of unitary \(N\times N\) matrices. It is proved that if \(\rho:G\to U(N)\) is a map, for which \[ \| \rho(fg)-\rho(f)\rho(g)\| ,\quad f,\,g\in G, \] is uniformly small enough, then \(\rho\) is a small perturbation of an ordinary unitary representation of \(G\) into \(U(N)\).