### Abstract

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) < ∞}.

Original language | English (US) |
---|---|

Pages (from-to) | 430-463 |

Number of pages | 34 |

Journal | Journal of Functional Analysis |

Volume | 146 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1997 |

### ASJC Scopus subject areas

- Analysis

## Fingerprint Dive into the research topics of 'Upper multiplicity and bounded trace ideals in C*-algebras'. Together they form a unique fingerprint.

## Cite this

*Journal of Functional Analysis*,

*146*(2), 430-463. https://doi.org/10.1006/jfan.1996.3041