On the symmetry of L1 algebras of locally compact motion groups,and the Wiener Tauberian theorem

Let G be a locally compact motion group, i.e., it is a semidirect product of a compact subgroup with a closed abelian normal subgroup, the action of the compact subgroup on the other one being by conjugation. The main result of this paper is that the group algebra of such a group is symmetric. This result is then used to prove that a generalization of the Wiener-Tauberian theorem holds for such groups. Precisely, it is shown that every proper closed two-sided ideal in L₁(G) is annihilated by an irreducible unitary representation of G, lifted to L₁(G).