Chapter 42 - Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups

Abstract

This chapter discusses some results and questions related to Chu duality and Bohr compactifications of locally compact groups. The problems presented in the chapter have appeared in the context of harmonic analysis on locally compact groups but they can be understood, and perhaps solved, adopting topological methods. This produces some genuine topological questions that can be handled using methods of harmonic analysis. Chu duality, called unitary duality by Chu, is based on giving a certain topological and algebraic structure to the set of finite dimensional representations of a topological group G. Denote to that end by Gxn the set of all continuous n-dimensional unitary representations of G. The set Gxn, equipped with the compact-open topology, is a locally compact space. The space Gx = ⊔n<w Gxn (as a topological sum) is called the Chu dual of G. The algebraic structure of Gx is given by two standard operations: the direct sum and the tensor product of representations, which are induced by the corresponding operations between finite dimensional operators. Basic definitions of Chu duality and the concepts of The Bohr compactification are also discussed in the chapter. It also focuses on determining to what extent the results concerning duality theory and Bohr topology of Abelian groups can be extended to the noncommutative context.