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

ISSN： 0025-5874

Source： Mathematische Zeitschrift, Vol.255, Iss.2, 2007-02, pp. : 245-282

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Abstract

We consider a properly converging sequence of non-characters in the dual space of a thread-like group and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if (_k) is a properly convergent sequence of non-characters in , then there is a trade-off between the number of limits  which are not characters, their degrees, and the strength of convergence i _ 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 (_k) is a properly converging sequence of non-characters in and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to ) 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 ) such that, for a in the Pedersen ideal of c ^*(G _N), exists (not identically zero) and is given by a sum of integrals over .