Upper multiplicity and bounded trace ideals in C*-algebras

R. J. Archbold, D. W B Somerset, John Spielberg

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

Upper and lower multiplicities MU(π, Ω) and ML(π, Ω) for an irreducible representation π of a C*-algebra A, relative to a net Ω = (πα) 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(πα(a)) ≤ ∑ MU(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 MU(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: MU(π) < ∞}.

Original languageEnglish (US)
Pages (from-to)430-463
Number of pages34
JournalJournal of Functional Analysis
Volume146
Issue number2
DOIs
StatePublished - Jun 1 1997

ASJC Scopus subject areas

  • Analysis

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