Three-Representation Problem in Banach Spaces

摘要

We provide the proof of a previously announced result that resolves the following problem posed by A. A. Kirillov. Let T be a presentation of a group G by bounded linear operators in a Banach space G and E. G be a closed invariant subspace. Then T generates in the natural way presentations T-1 in E and T2 in F := G/ E. What additional information is required besides T-1, T-2 to recover the presentation T? In finite-dimensional (and even in infinite dimensional Hilbert) case the solution is well known: one needs to supply a group cohomology class h is an element of H-1(G, Hom( F, E)). The same holds in the Banach case, if the subspace E is complemented in G. However, every Banach space that is not isomorphic to a Hilbert one has non-complemented subspaces, which aggravates the problem significantly and makes it non-trivial even in the case of a trivial group action, where it boils down to what is known as the threespace problem. This explains the title we have chosen. Asolution of the problem stated above has been announced by the author in 1976, but the complete proof, for nonmathematical reasons, has not been made available. This article contains the proof, as well as some related considerations of the functor Ext1 in the category Ban of Banach spaces.