Involutions on certain Banach algebras related to locally compact groups

‎Let $ L_0^\infty ({\frak G}‎, ‎1/\omega) $ be the space of all essentially bounded‎ ‎functions $ g $ on a locally compact group $ {\frak G} $ for which $ g/\omega $ vanishes‎ ‎at infinity‎, ‎where $ \omega $ is a weight function on $ {\frak G} $‎. ‎It is has recently‎ ‎shown that the dual space $ {L_0^\infty ({\frak G}‎, ‎1/\omega)}^* $ can be equipped with‎ ‎an Arens type product‎. ‎Here‎, ‎we show that the Banach algebra‎ ‎$ {L_0^\infty ({\frak G}‎, ‎1/\omega)}^* $ admits an involution if and only if $ {\frak G} $ is discrete‎.