Surjectivity of the comparison map in bounded cohomology for Hermitian Lie groups (Q2888918)

Let \(G\) be a connected semisimple Lie group and let \(\alpha\in H^\ast_c(G;{\mathbb R})\) be some class in the continuous cohomology over the reals. The question investigated in the paper under review is whether \(\alpha\) can always be represented by some bounded cocycle, i.e., whether the so-called comparison map \(H^\ast_{cb}(G;{\mathbb R}) \to H^\ast_c(G;{\mathbb R})\), which is induced by the inclusion map of bounded continuous functions, is surjective. This is an open question for more than 30 years. For some overview on this topic, see the proceedings article of \textit{N. Monod} [Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1183--1211 (2006; Zbl 1127.55002)]. The authors' main theorem states that if \(G\) has no compact factors and a finite center and each of its simple factors is either Hermitian or locally isomorphic to SO\(_0(p,q)\), \(pq\) even, or locally isomorphic to Sp\((p,q)\), \(p,q, \geq 1\), or an exceptional group with some Lie algebra listed by the author, then the comparison map is surjective. The main tool for proving this theorem is a universal map \(H^\ast(BG,{\mathbb R}) \to H_c^\ast(G,{\mathbb R})\) introduced by \textit{R. Bott} in 1972 [Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 161--170 (1975; Zbl 0306.57013)].