Banach halos and short isometries
Résumé
The aim of this article is twofold. First, we develop the notion of a Banach halo, similar to that of a Banach ring, except that the usual triangular inequality is replaced by the inequality $|a + b| \leq (|a| , |b|)_p$ involving the p-norm for some $p \in]0, +∞]$, or by the inequality $|a+b|\leq C\max(|a|,|b|)$. This allows us to have a flow of powers on Banach halos and to work, e.g., with the square of the usual absolute value on $\Z$. Then we define and study the group of short isometries of normed involutive coalgebras over a base commutative Banach halo. An aim of this theory is to define a representable group $K_n\subset GL_n$ whose points with values in $\R$ give $O_n(\R)$ and whose points with values in $\Q_p$ give $\GL_n(\Z_p)$, giving to the analogy between these two groups a kind of geometric explanation.
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