peripherally multiplicative operators between Banach function algebras

Let T : A —> B be a surjective operator between two Banach function algebras A and B with T 1 = 1. We show that if T satisfies the peripheral multiplicativity condition σ π (Tf.Tg) = σ π (f.g) for all f and g in A, where σ π (f) shows the peripheral spectrum of f, then T is a composition operator in modulus on the Silove boundary of A in the sense that /f (x)/= /Tf (t(x))/, for each f ∈ A and x ∈ ∂ (A) where t : ∂ (A) —> ∂ (B) is a homeomorphism between Silove boundaries of A and B.