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, M3(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. Introduction For any locally compact groupG, the convex, partially ordered semigroup P (G) of continuous positive definite functions on G, is a complete invariant of the group. In the discussion below, we show how P (G) can be used to recover the algebraic structure of G when G is finite or abelian. In the process we outline a duality theory for locally compact groups Throughout the paper, C∗(G) and L(G) will be defined as in [D], 13.9.1. For a C∗-algebra A, we will use A to denote the positive part of A, also as in [D]. The identity operator of A will be denoted by I and B(H) stands for the bounded linear operators on the Hilbert space H. For a, b ∈ H, the convex hull of a and b is written as co(a, b) and the orthogonal complement of the vector ξ ∈ H is written as ξ⊥. The notation Mn(A) denotes the n by n matrices with entries from A; e.g., A = C∗(G) and A = C are important examples used in this paper. The notation diag(λ1, ..., λn) will be used to denote a diagonal n by n matrix with λk in the k th diagonal entry. The symbol Zn will be used to denote the integers modulo n. 1. Binary Product and Order Structure Duality For any locally compact group G, the product structure of G is determined by the order structure of M3(C ∗(G)). The correspondence can be seen as a special case of the following theorem: 2000 Mathematics Subject Classification. Primary 43A35, 43A65 Secondary 43A40, 20D99, 06F99.