Banach halos and short isometries

Abstract

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 $|a+b|\le (|a|,|b|{)}_p$ involving the p-norm for some $p\in ]0,+\infty ]$, or by the inequality $|a+b|\le C\max (|a|,|b|)$. 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 $\mathbb{Z}$. 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 $K_n\subset {\mathrm{GL}}_n$ whose points with values in $\mathbb{R}$ give $O_n(\mathbb{R})$ and whose points with values in ${\mathbb{Q}}_p$ give GL${}_n({\mathbb{Z}}_p)$, giving to the analogy between these two groups a kind of geometric explanation.