## 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.

### MSC:

 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

Zbl 1010.14004
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