Involutions and trivolutions on second dual of algebras related to locally compact groups and topological semigroups

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation (*)-semigroup S when it is infinite non-discrete cancellative, (M_a(S)^{**}) does not admit an involution, and (M_a(S)^{**}) has a trivolution with range (M_a(S)) if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, (A(G)^{**}) always admits trivolution.