Three-representation problem

DOI10.1007/S11785-021-01079-6arXiv2006.07696MaRDI QIDQ6342839

Peter Kuchment

Publication date: 13 June 2020

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 $mathcalG$ by bounded linear operators in a Banach space $G$ and $EsubsetG$ 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 $hin{H}^1(mathcalG,Hom(F,E))$. 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 $Ex{t}^1$ in the category extbf{Ban} of Banach spaces.

Mathematics Subject Classification ID

Representations of general topological groups and semigroups (22A25) Invariant subspaces of linear operators (47A15) Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.) (46M18) Representation theory of linear operators (47A67)

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