Upper multiplicity and bounded trace ideals in C*-algebras

Upper and lower multiplicities M_U(π, Ω) and M_L(π, Ω) for an irreducible representation π of a C*-algebra A, relative to a net Ω = (π_α) in Â, 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 an upper semi-continuity result in [0, ∞] for trace functions on Â: lim sup Tr(π_α(a)) ≤ ∑ M_U(π, Ω) Tr(π(a)) (a ∈ A⁺), where the summation is taken over the cluster points of Ω. A characterization is given for the condition M_U(π, Ω) ≤ k, where k is a positive integer, from which it follows that a C*-algebra has all upper multiplicities finite if and only if it has bounded trace. More generally, the largest bounded trace ideal J of a C*-algebra A is given by Ĵ = {π ∈ Â: M_U(π) < ∞}.