A classification of the commutative Banach perfect semi-fields of characteristic 1. Applications

TL;DR

This paper introduces and analyzes commutative Banach perfect semi-fields of characteristic 1, establishing a Gelfand-Naimark type theorem that characterizes these structures as continuous functions on a spectrum, with various applications.

Contribution

It defines a new class of Banach semi-fields of characteristic 1 and proves a Gelfand-Naimark type theorem linking them to continuous functions on a spectrum.

Findings

Identification of Banach semi-fields with continuous functions on a spectrum

Development of a Gelfand-Naimark type theorem for these semi-fields

Multiple applications demonstrating the theory's utility

Abstract

We define and study the concept of commutative Banach perfect semi-field of characteristic 1 by using results from Connes-Consani. We prove a Gelfand-Naimark type theorem allowing to identify such a Banach semi-field F to the semi-field of all the continuous functions on a compact space which we define as the spectrum of F. We give many applications.