Banach $\widetilde{\mathbb C}$-algebras

TL;DR

This paper explores the structure and spectral theory of Banach algebras over Colombeau generalized complex numbers, highlighting differences from classical theory and proposing a specialized class of such algebras.

Contribution

It develops spectral theory for Banach $ ilde{C}$-algebras, identifies limitations of classical results, and introduces a class of algebras that mitigates these issues.

Findings

Classical Banach algebra properties do not fully extend to $ ilde{C}$-algebras.

A specific class of Banach $ ilde{C}$-algebras is proposed to overcome limitations.

Spectral theory is successfully developed for these generalized algebras.

Abstract

We study Banach $\widetilde{\mathbb C}$-algebras, i.e., complete ultra-pseudo-normed algebras over the ring $\widetilde{\mathbb C}$ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach $\widetilde{\mathbb C}$-algebras and indicate a particular class of Banach $\widetilde{\mathbb C}$-algebras that overcomes these limitations to a large extent. We also investigate C*-algebras over $\widetilde{\mathbb C}$.