The Eberlein Compactification of Locally Compact Groups

Abstract

A compact semigroup is, roughly, a semigroup compactification of a locally compact group if it contains a dense homomorphic image of the group. The theory of semigroup compactifications has been developed in connection with subalgebras of continuous bounded functions on locally compact groups.

The Eberlein algebra of a locally compact group is defined to be the uniform closure of its Fourier-Stieltjes algebra. In this thesis, we study the semigroup compactification associated with the Eberlein algebra. It is called the Eberlein compactification and it can be constructed as the spectrum of the Eberlein algebra.

The algebra of weakly almost periodic functions is one of the most important function spaces in the theory of topological semigroups. Both the weakly almost periodic functions and the associated weakly almost periodic compactification have been extensively studied since the 1930s. The Fourier-Stieltjes algebra, and hence its uniform closure, are subalgebras of the weakly almost periodic functions for any locally compact group. As a consequence, the Eberlein compactification is always a semitopological semigroup and a quotient of the weakly almost periodic compactification.

We aim to study the structure and complexity of the Eberlein compactifications. In particular, we prove that for certain Abelian groups, weak^{*}-closed subsemigroups of L^{\infty}[0, 1] may be realized as quotients of their Eberlein compactifications, thus showing that both the Eberlein and weakly almost periodic compactifications are large and complicated in these situations. Moreover, we establish various extension results for the Eberlein algebra and Eberlein compactification and observe that levels of complexity of these structures mimic those of the weakly almost periodic ones. Finally, we investigate the structure of the Eberlein compactification for a certain class of non-Abelian, Heisenberg type locally compact groups and show that aspects of the structure of the Eberlein compactification can be relatively simple.