Weak regularity of group algebras

A Banach algebra A is called weakly regular if its multiplicative semigroup is Einversive. We show that for a unimodular group G which admits an integrable unitary representation, L-1(G) is weakly regular. Moreover for a locally compact Abelian group, L-1(G) is weakly regular if and only if G is compact; while L-1(G) ** is weakly regular if and only if G is finite. All of our results hold, if we replace L-1(G) with M(G).