2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/104782Using a ``3 by 3 matrix trick'' we previously showed that multiplication in a C*-algebra A, an algebraic structure, is determined by the geometry of the C*-algebra of the 3 by 3 matrices with entries from A. As an application of this algebra-geometry duality we now construct an order theoretic based duality theory for all groups which are either locally compact abelian or finite. This construction generalizes the van Kampen--Pontriagin duality for locally compact abelian groups.Operator AlgebrasFunctional AnalysisGroup Theory54C40; 14E20text