An Explicit Duality for Finite Groups

Using 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.