Structure of the Eberlein compactification of locally compact Heisenberg type group $$\mathbb {Z}\times \mathbb {T} \times \mathbb {T}$$ Z × T × T

Abstract

Given a locally compact group G, the Eberlein compactification\(G^{e}\) is the spectrum of the uniform closure of the Fourier–Stieltjes algebra B(G). Hence, it is the semigroup compactification related to the unitary representations of G. \(G^e\) is a semitopological semigroup compactification and thus a quotient of the weakly almost periodic compactification of G. In this paper we aim to study the Eberlein compactification of the group \(\mathbb {Z} \times \mathbb {T} \times \mathbb {T}\) equipped with Heisenberg type multiplication. First, we will see that transitivity properties of the action of \(\mathbb {Z} \times \mathbb {T}\) on the central subgroup \(\mathbb {T}\) force some aspects of the structure of \((\mathbb {Z} \times \mathbb {T} \times \mathbb {T})^{e}\) to be quite simple. On the other hand, we will observe that the Eberlein compactification of the direct product group \(\mathbb {Z} \times \mathbb {T}\) is large with a complicated structure, and can be realized as a quotient of the Eberlein compactification \((\mathbb {Z} \times \mathbb {T} \times \mathbb {T})^{e}\).