On the support of the conjugation representation for solvable locally compact groups

The Glimm Ideal Space of a Two-Step Nilpotent Locally Compact Group

“…The proof of the following lemma imitates an argument from [20,Lemma 2.5], where it has been used to determine the support inĜ of the conjugation representation of G on L 2 (G) for a compactly generated, compact-free, two-step nilpotent group.…”

Locally compact groups whose conjugation representations satisfy a Kazhdan type property or have countable support

“…The difficulty in dealing with such questions for y G is that in general very little is known about suppy G . Although there are various classes of groups satisfying suppy G = (G/Z(G)Y (see [32] and [35]), even for finite groups or 2-step nilpotent simply connected Lie groups suppy G can be strictly contained in (G/Z(G))" (compare [19] and [20]).…”

“…For a locally compact group G with left Haar measure and modular function S the conjugation representation y G of G on L 2 (G) is defined by feL 2 (G), x,yeG. y G has been investigated recently (see [19,20,21,24,32,35]). For semi-simple Lie groups, a related representation has been studied in [25].…”

“…Clearly, y G is trivial on the centre Z(G) of G and separates the elements of G/Z(G). However, suppy G is generally strictly contained in (G/Z(G))" and intricate to determine (compare [19,20,32,35]). It is this unpleasant behaviour that raises most of the difficulties.…”

On 𝐶*-algebras associated to the conjugation representation of a locally compact group

“…However, so far it is much less understood than the left regular representation. The main difficulty arising is that, even for finite groups, the support of yG is generally strictly contained in the dual of G/Z(G) and is intricate to determine (compare [11], [12], [13], [20], and [22]). …”