Mathematische Zeitschrift

, Volume 255, Issue 2, pp 245–282 | Cite as

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

Article

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 (π k ) 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 i σ 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 (π k ) 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 (c k ) (with \(c_k\ne 0\)) such that, for a in the Pedersen ideal of C *(G N ), \(\lim_{k \rightarrow \infty} c_k {\rm Tr} ( \pi_k(a))\) exists (not identically zero) and is given by a sum of integrals over \(\mathbb{R}^{2}\).

Mathematics Subject Classification (2000)

Primary 22D10 Primary 22D25 Primary 22E27 Secondary 46L30 Secondary 43A40 

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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

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