Weak regularity of group algebras

A Banach algebra \(\mathcal {A}\) is called weakly regular if its multiplicative semigroup is E-inversive. 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).