Spectral measures over c-algebras of operators defined in $c_0$

TL;DR

This paper extends spectral theory to non-archimedean Banach algebras over c_0, establishing isometric isomorphisms with function spaces and defining spectral measures for operator analysis.

Contribution

It introduces a non-archimedean analogue of Gelfand spaces for certain Banach algebras and develops spectral measure theory in this context.

Findings

Algebras are isometrically isomorphic to spaces of continuous functions over compact sets.

Spectral measures are constructed that preserve projections.

Elements of the algebras can be represented as integrals with respect to these measures.

Abstract

The main goal of this work is to introduce an analogous in the non-archimedean context of the Gelfand spaces of certain Banach commutative algebras with unit. In order to do that, we study the spectrum of this algebras and we show that, under special conditions, these algebras are isometrically isomorphic to certain spaces of continuous functions defined over compacts. Such isometries preserve projections and allow to define associated measures which are known as spectral measures. We also show that each element of the algebras can be represented as an integral defined by these measures.