Mathematische Zeitschrift

, Volume 255, Issue 2, pp 245–282

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

Article

DOI: 10.1007/s00209-006-0023-1

Cite this article as:
Archbold, R.J., Ludwig, J. & Schlichting, G. Math. Z. (2007) 255: 245. doi:10.1007/s00209-006-0023-1

Abstract

We consider a properly converging sequence of non-characters in the dual space of a thread-like group \(G_N (N\geq {3})\) and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if (πk) is a properly convergent sequence of non-characters in \(\widehat{G}_N\), then there is a trade-off between the number of limits σ which are not characters, their degrees, and the strength of convergence iσ 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 (πk) is a properly converging sequence of non-characters in\(\widehat{G}_N\) and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to \(\mathbb{R}^{2}\)) 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 (ck) (with \(c_k\ne 0\)) such that, for a in the Pedersen ideal of C*(GN), \(\lim_{k \rightarrow \infty} c_k {\rm Tr} ( \pi_k(a))\) exists (not identically zero) and is given by a sum of integrals over \(\mathbb{R}^{2}\).

Mathematics Subject Classification (2000)

Primary 22D10Primary 22D25Primary 22E27Secondary 46L30Secondary 43A40

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Aberdeen, King’s CollegeScotlandUK
  2. 2.Département de Mathématiques, Île du SaulcyUniversité de MetzMetzFrance
  3. 3.Mathematisches InstitutTechnische Universität MünchenMünchenGermany