A characterization of certain weak∗-closed subalgebras of L∞(G)

AbstractLet G be an arbitrary locally compact Hausdorff group, and let L∞(G) be the set of essentially bounded measurable functions on G with respect to left invariant Haar measure. Let S be a linear subspace of L∞(G) which is (i) left and right translation invariant; (ii) weak∗-closed; (iii) self-adjoint, i.e., ƒ ϵ S implies ƒ ϵ S; and (iv) an algebra containing the constant functions. Then there exists a unique closed normal subgroup H of G such that S = {ƒ ϵ L∞(G): aƒ = ƒa = ƒ, ∀a ϵ H}. This extends to arbitrary locally compact groups, a result known only for Abelian groups