| Abstrakt: |
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:limsupTr(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)<∞}. |
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