*-Regularity of some classes of solvable groups

Some permanence properties of C∗-unique groups

“…By condition (U3) in Section 1, we have the following proposition which is an analogue of the corresponding result for Ã-regular groups in [6,Corollary 1]. In fact, it can be shown that under the assumption of this proposition, for any injective nondegenerate representation p of L 1 ðGÞ and any f AC c ðGÞ; one has jpð f Þj ¼ j f j C Ã ðGÞ note that the support of f is a subset of a compactly generated open subgroup of G).…”

“…Example 2.11. (a) The non-C Ã -unique group given by Poguntke [24] is a semi-direct product of two C Ã -unique groups (in fact, both are semi-direct products of abelian groups; see [6,Theorem 3]). …”

The C*-algebra of the variable Mautner group

“…It had been shown in [3] that every connected locally compact group G of polynomial group (in particular the group M θ ) is * -regular. This means that the group algebra L 1 (G) is star regular.…”

The C*-algebra of Boidol's group

“…In this way, the C*-algebra of every exponential Lie group of dimension less than or equal to 4 has been explicitly determined with one exception, namely Boidol's group G, which is an extension of the Heisenberg group by the reals with the roots 1 and −1. This group can be considered to be wild with respect to its group algebra L 1 (G), since it is not star-regular (see [2,5]), is not hermitian [3] and it does not have the Wiener property (see [15]) as all the other simply connected solvable Lie groups of dimension less than or equal to 4 do.…”