Note: in this post, every topological group under consideration is assumed to be Hausdorff.
Given a locally compact abelian group, one can construct its dual group, i.e. its group of (unitary) characters. This turns out to be a locally compact abelian group with respect to the compact-open topology. Pontryagin duality then tells us that there is a natural isomorphism of topological groups between the original group and its double dual. Thus, if we know only the unitary characters of the group (along with their algebraic and topological structure), we can recover the original group (up to isomorphism) by taking its dual.
Similarly, given the irreducible unitary representations of a compact group, one can construct a compact group out of them which turns out to be isomorphic to the original group (the Tannaka duality theorem). Thus the irreducible unitary representations of a compact group contain enough information to recover the original group.
How much of this remains true for general locally compact groups? By the Gelfand-Raikov theorem, such groups have many irreducible unitary representations (enough to separate points). The question is: can one associate a canonical group structure and/or a topological structure to the irreducible representations of the group (or their equivalence classes) so that one recovers the original group (along with its topology) up to isomorphism?
If the answer is "no", can this be remedied by considering more general representations of the group, e.g. representations which aren't necessarily unitary?
I'm not sure if these questions have an easy answer, so I should mention that I'm mostly interested in the cases where the group is a connected nilpotent Lie group or a connected semisimple Lie group.