Three-representation problem

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 \textbf{Ban} of Banach spaces.