### 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 |

State | Published - Jun 1 1997 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*146*(2), 430-463.

**Upper multiplicity and bounded trace ideals in C*-algebras.** / Archbold, R. J.; Somerset, D. W B; Spielberg, John.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 146, no. 2, pp. 430-463.

}

TY - JOUR

T1 - Upper multiplicity and bounded trace ideals in C*-algebras

AU - Archbold, R. J.

AU - Somerset, D. W B

AU - Spielberg, John

PY - 1997/6/1

Y1 - 1997/6/1

N2 - 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) < ∞}.

AB - 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) < ∞}.

UR - http://www.scopus.com/inward/record.url?scp=0031168093&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031168093&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031168093

VL - 146

SP - 430

EP - 463

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -