Chain-center duality for locally compact groups

The chain group Ch(G) of a locally compact group G has one generator (Formula presented.) for each irreducible unitary G-representation ρ, a relation (Formula presented.) whenever ρ is weakly contained in (Formula presented.), and (Formula presented.) for the representation (Formula presented.) contragredient to ρ. G satisfies chain-center duality if assigning to each (Formula presented.) the central character of ρ is an isomorphism of Ch(G) onto the dual (Formula presented.) of the center of G. We prove that G satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. Müger’s result compact groups satisfy chain-center duality.