Duality for Abelian Groups

Abstract

In this chapter we are mainly interested in the study of abelian locally compact groups A, their dual groups $\hat{A}$ together with various associated group algebras. Using the Gelfand-Naimark Theorem as a tool, we shall then give a proof of the Plancherel Theorem, which asserts that the Fourier transform extends to a unitary equivalence of the Hilbert spaces $L^2(A)$ and $L^2(\widehat A)$. We also prove the Pontryagin Duality Theorem that gives a canonical isomorphism between A and its bidual ${\hat{\hat{A}}}$.