R�ume primitiver Ideale von Gruppenalgebren

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

“…Part (ii) of the next lemma shows that Definition 1.2 is equivalent to Boidol's original definition, a characterization which has already been proved in [4]. (ii) A Banach * -algebra A is * -regular if and only if Ψ is a homeomorphism with respect to the Jacobson topologies on Prim C * (A) and Prim * A.…”

“…Here are a few examples of * -regular Banach * -algebras: If G is a connected locally compact group such that its Haar measure has polynomial growth, then G is * -regular. Boidol proved this fact in Theorem 2 of [4] based on ideas of Dixmier in [7]. Jenkins has shown in Theorem 1.4 of [15] that connected nilpotent Lie groups have polynomial growth.…”

Functional calculus and *-regularity of a class of Banach algebras

“…We now recall the following result from [2]. As we are in a more general setting, we repeat their argument here for clarity.…”

Unitary representations of diamond groups

“…With some additional effort one can show that all our results in the previous and in the following sections remain true if we stay with L 1 -covariance algebras and if instead of a separable commutative C Ã -algebra b we consider a separable commutative symmetric regular Banach Ã-algebra b with a bounded approximate identity. A little more precisely, all the L 1 -versions of the C Ã -algebras we are going to investigate are Ã-regular in the sense of [3], i.e., their spaces of kernels of irreducible involutive representations endowed with the Jacobson topology are homeomorphic to the primitive ideal space of their C Ã -completions. And for the algebra we studied already, namely L 1 U n N, this is equally true because clearly U n N has a polynomially growing Haar measure, see [3].…”

“…A little more precisely, all the L 1 -versions of the C Ã -algebras we are going to investigate are Ã-regular in the sense of [3], i.e., their spaces of kernels of irreducible involutive representations endowed with the Jacobson topology are homeomorphic to the primitive ideal space of their C Ã -completions. And for the algebra we studied already, namely L 1 U n N, this is equally true because clearly U n N has a polynomially growing Haar measure, see [3].…”