El Teorema de Gelfand Naimark desde una perspectiva Categ\'orica The Gelfand--Naimark Theorem from a Categorical Perspective

TL;DR

This paper demonstrates a categorical equivalence between Hausdorff-Compact topological spaces and unital commutative $C^*$-algebras, reinterpreting the Gelfand-Naimark Theorem in categorical terms and exploring its applications.

Contribution

It establishes a categorical equivalence between topological spaces and $C^*$-algebras, providing a new perspective on the Gelfand-Naimark Theorem and its applications.

Findings

Categorical equivalence between topological spaces and $C^*$-algebras.

Reinterpretation of Gelfand-Naimark Theorem in categorical language.

Applications of the categorical perspective to typical problems.

Abstract

Este art\'iculo presenta como resultado principal la equivalencia entre, las categor\'ias de espacios topol\'ogicos Hausdorff-Compactos y la categor\'ia de las $C^*-$\'algebras conmutativas con unidad, producto de la ``traducci\'on'' en este lenguaje del teorema de Gelfand--Naimark presentado en 1943. Haremos un recorrido sobre las principales ideas del an\'alisis y el \'algebra, conjugadas con \'exito, en el estudio de la teor\'ia de \'Algebras de Banach. As\'i mismo estableceremos, a forma de conclusi\'on, diversas aplicaciones que resultan naturalmente posibles a la luz de la ``analog\'ia y generalizaci\'on'' que nos permiten la teor\'ia de categor\'ias. Palabras claves: $C^*$-algebras, Categor\'ias, Espacios Topol\'ogicos, Teorema de Gelfand-Naimark, Teor\'ia de Representaciones. The goal of this paper is to prove the categorical equivalence between the category of Hausdorff-Compact topological spaces and the category of Unital Commutative $C^*$-algebras. This equivalence can be interpreted as a way of rewriting the well known Gelfand-Naimark Theorem in a categorical language. We will present the basic concepts in the theory of Banach Algebras as a successful link between Analysis and Algebra. Likewise, we will show some applications due to this new perspective, highlighting the categorical connection through proofs of typical problems that don't have an easy solution in $C^*-$algebra. Keywords: Category Theory, $C^*$-algebras, Gelfand-Naimark Theorem, Topological Spaces, Representation Theory.