Involutions on certain Banach algebras related to locally compact groups

Let (mathcal{{A}}) be a Banach algebra and let (mathcal{{X}}) be an introverted closed subspace of (mathcal{{A}}^*) . Here, we give necessary and sufficient conditions for that the dual algebra (mathcal{{X}}^*) of (mathcal{{X}}) or the topological centers ({mathfrak {Z}}_t^{(1)}(mathcal{{X}}^{*})) and ({mathfrak {Z}}_t^{(2)}(mathcal{{X}}^{*})) of (mathcal{{X}}^*) are Banach (*) -algebras. We finally apply these results to the Banach space (L_0^infty (G)) of all equivalence classes of essentially bounded functions vanishing at infinity on a locally compact group (G) .