Three-Representation Problem in Banach Spaces

Abstract

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 $$\mathcal {G}$$ by bounded linear operators in a Banach space G and $$E\subset G$$ be a closed invariant subspace. Then T generates in the natural way presentations $$T_1$$ in E and $$T_2$$ 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 $$h\in H^1(\mathcal {G},Hom(F,E))$$ . 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 three-space problem. This explains the title we have chosen. A solution of the problem stated above has been announced by the author in 1976, but the complete proof, for non-mathematical reasons, has not been made available. This article contains the proof, as well as some related considerations of the functor $$Ext^1$$ in the category Ban of Banach spaces.