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

The famous van der Waerden theorem [B.L. van der Waerden, Stetigkeitssatze fur halbeinfache Liesche Gruppen, Math. Z. 36 (1933) 780-786] concerns the continuity of finite-dimensional representations of compact semisimple Lie groups. It turns out that this theorem is stable, i.e., if G is a compact semisimple Lie group and rho : G -> U(N) is a mapping for which the norm parallel to rho(gg') - rho(g)rho(g')parallel to, g, g' epsilon G, is uniformly small enough, then the mapping rho is a small perturbation of a (necessarily continuous) ordinary unitary representation of G into U(N). This completely answers the question attributed to Milman by Kazhdan [D. Kazhdan, On epsilon-representations, Israel J. Math. 43 (4) (1982) 315-323] and gives a partial answer to Gromov's question [M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional Analysis on the Eve of the 21st Century, vol. II, Dordrecht, Boston, MA, 1996, pp. 1-213]. (C) 2006 Elsevier Inc. All rights reserved.