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

Abstract

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 · q_ǀn is closed in n^∗. We then show that for every D-orbit Ω in g^∗ the kernel ker_C^∗(Ω) of Ω in the C^∗-algebra of G is L¹-determined, which means that ker_C^∗(Ω) is the closure of the kernel ker_L¹(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 ker_C^∗(π_O) in C^∗(G) is not L¹-determined.