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

Robert J Archbold, J. Ludwig, G. Schlichting

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 languageEnglish
Pages (from-to)245-282
Number of pages38
JournalMathematische Zeitschrift
Volume255
Issue number2
Early online date11 Jul 2006
DOIs
Publication statusPublished - 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

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

In: Mathematische Zeitschrift, Vol. 255, No. 2, 02.2007, p. 245-282.

Research output: Contribution to journalArticle

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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.

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