*-Regularity of exponential Lie groups

L 1-determined ideals in group algebras of exponential Lie groups

“…Jenkins has shown in Theorem 1.4 of [15] that connected nilpotent Lie groups have polynomial growth. If G is a metabelian connected locally compact group, then G is * -regular, see Theorem 3.5 of [2]. Moreover the following is true: If G is a compactly generated, locally compact group with polynomial growth and if w is a symmetric weight function on G which satisfies the non-abelian-Beurling-Domar condition (BDna) of [10], then L 1 (G, w) is * -regular.…”

“…They can be found in Boidol's paper [2], and in a more general context in [3]. For the convenience of the reader we give a short proof for the if-part of Theorem 5.4 of [2] using the results of the previous section. The following definition has been adapted from the introduction of [3].…”

“…In the introduction of [2] Boidol defined * -regularity of Banach * -algebras. We restate his definition and add the notion of primitive * -regularity.…”

On the Dual of an Exponential Solvable Lie Group

“…Brown in [3] that if G is nilpotent, n is a homeomorphism. K. Joy in a later paper [7] gives a much shorter proof of Brown's Theorem using results of J. M. G. Fell pertaining to the space S(G) of subgroup representation pairs (7r, H), where H is a closed connected subgroup of G and ir is an unitary equivalence class of representations of H. Two results on the bicontinuity of n when G is exponential are due to J. Boidol [2] and H. Fujiwara [6]. Boidol shows that n~l is continuous provided that G is *-regular; *-regularity is seen to fail however even for a completely solvable group of dimension four.…”

Sur la bicontinuité de l'application de Dixmier pour les algèbres de Lie résolubles

“…[2] et [6] pour les concepts généraux utilisés. 1 g) (resp. g)) le plus grand idéal bilatère de U g) contenu dans U( g).…”