ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 22, 18, 20 Shtern, Alexander, Moscow State University, Russia Quasi-Symmetry and Amenability A map T S → L(H) is a (C, ε)-quasi-representation (qrn) of a group G in a Hilbert space H if ∥T (gh) − T (g)T (h)∥ ≤ ε, g, h ∈ G, and ∥T (g)∥, ∥T (g)^{−1}∥ ≤ C, g ∈ G. A group G is said to be stably representable if, for any real C ≥ 1 and ε > 0, there exists δ > 0 such that, for any (C, ε)-qrn of G in a Hilbert space, there exists a representation R of G in the same space with ∥T (g) − R(g)∥ ≤ ε and ∥T (g)^{−1} − R(g)^{−1}∥ ≤ ε, g ∈ G. It is proved that if an almost connected locally compact group is stably representable, then it is amenable (the converse follows from a result of B. E. Johnson), and related topics on the second real bounded cohomology of connected Lie groups are discussed.