On the ${\Ext}^2$-problem for Hilbert spaces

Sánchez, Félix Cabello; Castillo, Jesús M. F.; Corrêa, Willian H. G.; Ferenczi, Valentin; Garcia, Ricardo Alexandrino

doi:10.48550/arxiv.1908.06529

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“…The way it was defined, Ext 1 (E, F ) was just a set. However, one can equip it with a vector space structure such that Ext 1 becomes a bi-functor from the category Ban of Banach spaces with bounded linear operators as morphisms 7 , into the category of vector spaces with linear mappings as morphisms. Namely, Theorem 1.7 enables one to transfer linear structure from S(F, E) onto Ext 1 (F, E), thus turning the latter into a vector space.…”

Section: Definition 13 the Set Of Equivalence Classes Of Extensions Ofmentioning

confidence: 99%

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Three-representation problem

Kuchment

2020

Preprint

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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 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 ∈ 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 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.

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Section: Definition 13 the Set Of Equivalence Classes Of Extensions Ofmentioning

confidence: 99%

“…Modulo checking functoriality 7. Higher Ext i functors can be also defined for any natural number i (see, e.g.,[18,19,43]).…”

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confidence: 99%