L1-DETERMINED PRIMITIVE IDEALS IN THE C∗-ALGEBRA OF AN EXPONENTIAL LIE GROUP WITH CLOSED NON-∗-REGULAR ORBITS

Let G = exp(g) be an exponential solvable Lie group and Ad(G) ⊂ D an exponential solvable Lie group of automorphisms of G. Assume that for every non-∗-regular orbit D · q, q ∈ g∗, of D = exp(∂) in g∗, there exists a nilpotent ideal n of g containing ∂ · g such that D · qǀn is closed in n∗. We then show that for every D-orbit Ω in g∗ the kernel kerC∗(Ω) of Ω in the C∗-algebra of G is L1-determined, which means that kerC∗(Ω) is the closure of the kernel kerL1(Ω) of Ω in the group algebra L1(G). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group D = Ad(G). We finally give an example of a non-closed non-∗-regular orbit of an exponential solvable group G and of a coadjoint orbit O ⊂ g∗, for which the corresponding kernel kerC∗(πO) in C∗(G) is not L1-determined.