  The chain group $C(G)$ of a locally compact group $G$ has one generator
$g_{\rho}$ for each irreducible unitary $G$-representation $\rho$, a relation
$g_{\rho}=g_{\rho'}g_{\rho"}$ whenever $\rho$ is weakly contained in
$\rho'\otimes \rho"$, and $g_{\rho^*}=g_{\rho}^{-1}$ for the representation
$\rho^*$ contragredient to $\rho$. $G$ satisfies chain-center duality if
assigning to each $g_{\rho}$ the central character of $\rho$ is an isomorphism
of $C(G)$ onto the dual $\widehat{Z(G)}$ 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. M\"{u}ger's result compact
groups satisfy chain-center duality.
