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(d) in g, there exists a nilpotent ideal n of g containing d · g such that D · q|n is closed in n. We then show that for every D-orbit  in g the kernel kerC∗() of  in the C-algebra of G is L1-determined, which means that kerC∗() is the closure of the kernel kerL1() of  in the group algebra L 1(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.