Análisis armónico algebraico.

Abstract

[ES]El objetivo de este trabajo es sustraer del Análisis Armónico estándar aquello más elemental y natural y ver en qué sentido es universal y, por tanto, extrapolable a la situación algebraica. En efecto, en análisis armónico se considera un grupo (continuo y abeliano) y se estudia su álgebra de medidas (o dual continuo de su espacio de funciones) con su producto natural de convolución. Esta se interpreta como álgebra de funciones sobre un grupo denominado grupo dual (su espectro, esto es: el grupo de caracteres o de representaciones simples). En este momento es esencial el ver hasta qué punto este proceso de paso al dual es reflexivo, pues ello daría que el dual del grupo dual es el grupo de partida y, por tanto, se tendría una dualidad completa (una anti equivalencia)

[EN]The aim of this work is to subtract from the standard Harmonic Analysis that which is most elementary and natural, and to see in what sense it is universal and therefore and see in what sense it is universal and, therefore, extrapolable to the algebraic situation. In fact, in harmonic analysis, we consider a group (continuous and abelian) and we study its algebra of measures (or continuous dual of its space of functions) with its natural convolution product. This is interpreted as a function algebra on a group called the dual group (its spectrum, i.e. the group of characters or simple representations). At this point, it is essential to see to what extent this process of passage to the dual is reflexive, since this would give that the dual of the dual group is the starting group and, therefore, one would have a complete duality (an anti-equivalence) between the geometry of the group and the geometry of the dual group.