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

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 $\mathbb{R}^{2}$.