Structure of the Eberlein compactification of locally compact Heisenberg type group ZxTxT

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 ZxTxT equipped with Heisenberg type multiplication. First, we will see that transitivity properties of the action of ZxT on the central subgroup T force some aspects of the structure of (ZxTxT) to be quite simple. On the other hand, we will observe that the Eberlein compactification of the direct product group ZxT is large with a complicated structure, and can be realized as a quotient of the Eberlein compactification (ZxTxT)(e).