Duality for Locally Compact Abelian Groups

Abstract

For a locally compact abelian group G, its group Ĝ of characters (i.e., continuous homomorphisms from G to S ¹) also acquires the structure of a topological group. In this chapter, we give two distinctive characterizations of what turns out to be the same underlying topology for Ĝ and examine this topology in detail. The main result is the Pontryagin duality theorem, which says in effect that G and Ĝ are mutually dual, both algebraically and topologically. To prove this, we build upon the results of the previous chapter, especially insofar as the introduction of functions of positive type makes a critical correspondence with the theory of unitary representations.