### Abstract

We consider a properly converging sequence of non-characters in the dual space of a thread-like group G(N)(N >= 3) and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if ( pi(k)) is a properly convergent sequence of non-characters in (G) over cap (N), then there is a trade-off between the number of limits sigma which are not characters, their degrees, and the strength of convergence is to each of these limits ( Theorem 3.2). This enables us to describe various possibilities for maximal limit sets consisting entirely of non-characters ( Theorem 4.6). In Sect. 5, we show that if (pi(k)) is a properly converging sequence of non-characters in (G) over cap (N) and if the limit set contains a character then the intersection of the set of characters ( which is homeomorphic to R-2) with the limit set has at most two components. In the case of two components, each is a half-plane. In Theorem 7.7, we show that if a sequence has a character as a cluster point then, by passing to a properly convergent subsequence and then a further subsequence, it is possible to find a real null sequence (c(k)) ( with ck not equal 0) such that, for a in the Pedersen ideal of C*(G(N)), lim(k ->infinity) c(k)Tr(pi(k)(a)) exists (not identically zero) and is given by a sum of integrals over R-2.

Original language | English |
---|---|

Pages (from-to) | 245-282 |

Number of pages | 38 |

Journal | Mathematische Zeitschrift |

Volume | 255 |

Issue number | 2 |

Early online date | 11 Jul 2006 |

DOIs | |

Publication status | Published - Feb 2007 |

### Keywords

- primary 22D10
- primary 22D25
- primary 22E27
- secondary 46L30
- secondary 43A40
- C-asterisk-algebras
- locally compact-groups
- irreducible representations
- lower multiplicity
- bounded trace
- topology

### Cite this

*Mathematische Zeitschrift*,

*255*(2), 245-282. https://doi.org/10.1007/s00209-006-0023-1

**Limit sets and strengths of convergence for sequences in the duals of thread-like Lie goups.** / Archbold, Robert J; Ludwig, J.; Schlichting, G.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 255, no. 2, pp. 245-282. https://doi.org/10.1007/s00209-006-0023-1

}

TY - JOUR

T1 - Limit sets and strengths of convergence for sequences in the duals of thread-like Lie goups

AU - Archbold, Robert J

AU - Ludwig, J.

AU - Schlichting, G.

PY - 2007/2

Y1 - 2007/2

N2 - We consider a properly converging sequence of non-characters in the dual space of a thread-like group G(N)(N >= 3) and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if ( pi(k)) is a properly convergent sequence of non-characters in (G) over cap (N), then there is a trade-off between the number of limits sigma which are not characters, their degrees, and the strength of convergence is to each of these limits ( Theorem 3.2). This enables us to describe various possibilities for maximal limit sets consisting entirely of non-characters ( Theorem 4.6). In Sect. 5, we show that if (pi(k)) is a properly converging sequence of non-characters in (G) over cap (N) and if the limit set contains a character then the intersection of the set of characters ( which is homeomorphic to R-2) with the limit set has at most two components. In the case of two components, each is a half-plane. In Theorem 7.7, we show that if a sequence has a character as a cluster point then, by passing to a properly convergent subsequence and then a further subsequence, it is possible to find a real null sequence (c(k)) ( with ck not equal 0) such that, for a in the Pedersen ideal of C*(G(N)), lim(k ->infinity) c(k)Tr(pi(k)(a)) exists (not identically zero) and is given by a sum of integrals over R-2.

AB - We consider a properly converging sequence of non-characters in the dual space of a thread-like group G(N)(N >= 3) and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if ( pi(k)) is a properly convergent sequence of non-characters in (G) over cap (N), then there is a trade-off between the number of limits sigma which are not characters, their degrees, and the strength of convergence is to each of these limits ( Theorem 3.2). This enables us to describe various possibilities for maximal limit sets consisting entirely of non-characters ( Theorem 4.6). In Sect. 5, we show that if (pi(k)) is a properly converging sequence of non-characters in (G) over cap (N) and if the limit set contains a character then the intersection of the set of characters ( which is homeomorphic to R-2) with the limit set has at most two components. In the case of two components, each is a half-plane. In Theorem 7.7, we show that if a sequence has a character as a cluster point then, by passing to a properly convergent subsequence and then a further subsequence, it is possible to find a real null sequence (c(k)) ( with ck not equal 0) such that, for a in the Pedersen ideal of C*(G(N)), lim(k ->infinity) c(k)Tr(pi(k)(a)) exists (not identically zero) and is given by a sum of integrals over R-2.

KW - primary 22D10

KW - primary 22D25

KW - primary 22E27

KW - secondary 46L30

KW - secondary 43A40

KW - C-asterisk-algebras

KW - locally compact-groups

KW - irreducible representations

KW - lower multiplicity

KW - bounded trace

KW - topology

U2 - 10.1007/s00209-006-0023-1

DO - 10.1007/s00209-006-0023-1

M3 - Article

VL - 255

SP - 245

EP - 282

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 2

ER -