Strength of convergence in duals of C*-algebras and nilpotent Lie groups

By using trace formulae, the recent concept of upper multiplicity for an irreducible representation of a C*-algebra is linked to the earlier notion of strength of convergence in the dual of a nilpotent Lie group G. In particular, it is shown that if π∈Ĝ has finite upper multiplicity then this integer is the greatest strength with which a sequence in Ĝ can converge to π. Upper multiplicities are calculated for all irreducible representations of the groups in the threadlike generalization of the Heisenberg group. The values are computed by combining new C*-theoretic results with detailed analysis of the convergence of coadjoint orbits and they show that every positive integer occurs for this class of groups.