Banach halos and short isometries

TL;DR

This paper introduces Banach halos, a generalization of Banach rings with modified triangle inequalities, and studies their short isometries to unify classical groups like orthogonal and p-adic groups under a geometric framework.

Contribution

The paper develops the concept of Banach halos with alternative inequalities and defines a group of short isometries, providing a geometric perspective linking real and p-adic classical groups.

Findings

Defined Banach halos with p-norm based inequalities.

Constructed a group of short isometries for normed involutive coalgebras.

Proposed a representable group linking O_n(R) and GL_n(Z_p).

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 $|a + b| \leq (|a| , |b|)_p$ involving the p-norm for some $p \in]0, +\infty]$, 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 $\mathbb{Z}$. 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 $K_n\subset {\rm GL}_n$ 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.