Groups with few finite dimensional unitary representations

The Bohr compactification of an arbitrary topological group G is defined as the group compactification (bG, b) with the following universal property: for every continuous homomorphism h from G into a compact group K there is a continuous homomorphism h^b from bG into K extending h in the sense that h=hᵇ b. The Bohr compactification (bG, b) is the unique (up to equivalence) largest compactification of G. Although, for locally compact Abelian groups, the Bohr compactification is a big monster, for non-Abelian groups the situation is much more interesting and it can be said that all options are possible. Here we are interested in locally compact groups whose Bohr compactification is small. Among other results, we characterize when the Bohr the Bohr compactification of a locally compact group is topologically isomorphic to its Chu or unitary quasi-dual. Our results fixe some incorrect statements appeared in the literature.