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

Abstract

A locally compact group G is said to be ∗-regular if the natural map Ψ : Prim C*(G)  Prim*L1(G) is a homeomorphism with respect to the Jacobson topologies on the primitive ideal spaces Prim C*(G) and Prim*L1(G). In 1980 J. Boidol characterized the ∗-regular ones among all exponential Lie groups by a purely algebraic condition. In this article we introduce the notion of L1-determined ideals in order to discuss the weaker property of primitive ∗-regularity. We give two sufficient criteria for closed ideals I of C*(G) to be L1-determined. Herefrom we deduce a strategy to prove that a given exponential Lie group is primitive ∗-regular. The author proved in his thesis that all exponential Lie groups of dimension ≤ 7 have this property. So far no counter-example is known. Here we discuss the example G = B5, the only critical one in dimension ≤ 5.