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Espaces de Banach de dimension finie. (Finite-dimensional Banach Spaces). (French) Zbl 1044.11102
The paper gives a formalization of some constructions contained in the proof of the equivalence of the categories of weakly admissible and admissible filtered \((\varphi,N)\)-modules [P. Colmez and J.-M. Fontaine, Invent. Math. 140, No. 1, 1–43 (2000; Zbl 1010.14004)].
Let \(K\) be a field of characteristic zero complete with respect to some discrete valuation, \(\overline{K}\) be its algebraic closure, and \(C\) be the completion of \(\overline{K}\). First the author defines a class of sympathetic Banach algebras over \(C\) with spectral norms, possessing some narural properties. A Banach Space (with a capital “S”) is defined as a covariant functor from the category of sympathetic Banach algebras into the category of Banach spaces over \(\mathbb Q_p\). A Banach Space in the above sense is said to be of a finite dimension if the space assigned to an algebra \(A\) differs from \(A^d\), \(d<\infty\), roughly speaking, by a finite dimensional vector space over \(\mathbb Q_p\). Such objects form a category whose properties are studied in detail.
The author discusses relations with a variety of subjects including filtered \((\varphi,N)\)-modules, semi-stable \(p\)-adic representations of \(\text{Gal} (\overline K/K)\), \(p\)-adic periods etc. Another related subject is the study of a “giant” skew field whose center is \(\mathbb Q_p\) while \(C\) is its maximal commutative subfield.

11S85 Other nonanalytic theory
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
14G20 Local ground fields in algebraic geometry
46B99 Normed linear spaces and Banach spaces; Banach lattices
46H99 Topological algebras, normed rings and algebras, Banach algebras
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