Constructive Theory of Banach algebras

TL;DR

This paper develops a constructive framework for Banach algebra theory, providing elementary proofs of classical results and exploring a localic perspective on spectra, inspired by foundational mathematicians.

Contribution

It introduces a constructive approach to Banach algebras, offering new proofs and a novel localic perspective on spectra, extending classical results.

Findings

Elementary proofs of Wiener's inverse Fourier series theorem

Elementary proofs of Wiener's Tauberian theorem

A foundation for localic spectrum analysis

Abstract

We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra.