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 pi is an element of (G) over cap has finite upper multiplicity then this integer is the greatest strength with which a sequence in (G) over cap can converge to pi. 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. (C) 2001 Academic Press.