PERIPHERALLY MULTIPLICATIVE OPERATORS ON UNITAL COMMUTATIVE BANACH ALGEBRAS

Let T : A -> B be a surjective operator between two unital semisimple commutative Banach algebras A and B with T1 = 1. We show that if T satisfies the peripheral multiplicativity condition sigma(pi) (Tf.Tg) - sigma(pi) (f.g) for all f and g in A, where sigma(pi) (f) shows the peripheral spectrum of f, then T is a composition operator in modulus on the. Silov boundary of A in the sense that vertical bar f(x)vertical bar = vertical bar Tf(tau(x))vertical bar, for each f is an element of A and x is an element of partial derivative (A) where tau : partial derivative(A) -> partial derivative (B) is a homeomorphism between Silov boundaries of A and B.