A Corollary to a Triviality Theorem for Quasirepresentations of an Amenable Group in Reflexive Banach Spaces

As is known, for a sufficiently small defect of a (not necessarily bounded) quasirepresentation of an amenable group in a reflexive Banach space \(E\) with dense set of bounded orbits, there is an extension of this quasirepresentation for which there is a close ordinary representation of the group in the space of this extension. In the present note it is proved that, if the original quasirepresentation \(\pi\) of an amenable group \(G\) in a reflexive Banach space \(E\) is a pseudorepresentation, then an ordinary representation of \(G\), in the vector subspace \(L\) of \(E\) formed by vectors with bounded orbits and equipped with a natural Banach norm, which is close to \(\pi|_L\) (this ordinary representation exists if the defect of \(\pi\) is sufficiently small) is equivalent to \(\pi|_L\).