, Volume 255, Issue 2, pp 245-282
Date: 11 Jul 2006

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

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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 \(\mathbb{R}^{2}\) .