Upper Multiplicity and Bounded Trace Ideals inC*-Algebras

Upper and lower multiplicitiesMU(π, Ω) andML(π, Ω) for an irreducible representationπof aC*-algebraA, relative to a netΩ=(πα) inÂ, are shown to generalize the multiplicity numbers obtained by previous authors in trace formulae for (group)C*-algebras. This leads, in the presence of an auxiliary finiteness condition, to anuppersemi-continuity result in [0, ∞] for trace functions onÂ:lim sup Tr(πα(a))⩽∑ MU(π, Ω) Tr(π(a)) (a∈A+),where the summation is taken over the cluster points ofΩ. A characterization is given for the conditionMU(π, Ω)⩽k, wherekis a positive integer, from which it follows that aC*-algebra has all upper multiplicities finite if and only if it has bounded trace. More generally, the largest bounded trace idealJof aC*-algebraAis given byĴ={π∈Â: MU(π)<∞}.