TY - JOUR

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

AU - Archbold, R. 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 GN (N≥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 ĜN, 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 ĜN and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to Rdbl; ) 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 κ≠ 0) such that, for a in the Pedersen ideal of C *(G N), lim k→∞exists (not identically zero) and is given by a sum of integrals over ℝ.

AB - We consider a properly converging sequence of non-characters in the dual space of a thread-like group GN (N≥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 ĜN, 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 ĜN and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to Rdbl; ) 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 κ≠ 0) such that, for a in the Pedersen ideal of C *(G N), lim k→∞exists (not identically zero) and is given by a sum of integrals over ℝ.

UR - http://www.scopus.com/inward/record.url?scp=33751503371&partnerID=8YFLogxK

U2 - 10.1007/s00209-006-0023-1

DO - 10.1007/s00209-006-0023-1

M3 - Article

AN - SCOPUS:33751503371

SN - 0025-5874

VL - 255

SP - 245

EP - 282

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

ER -