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Abstract. Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if/.g = 0 implies Tf.Tg = 0, for all f,g e A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.Finally, we show that linear isometries between regular uniform algebras are separating. As corollaries, a classical theorem of Nagasawa ([19]) and the Banach-Stone theorem (both for regular uniform algebras) are easily inferred. Introduction.The basic problem in the theory of automatic continuity of linear operators consists of giving algebraic conditions on two Banach algebras si and 38 which ensure that every algebra homomorphism !T:s4-*2ft is necessarily continuous. The most important positive result in this context is due to B. E. Johnson ([15]). Every algebra homomorphism of a Banach algebra onto a semisimple Banach algebra is continuous.In this paper we shall study the automatic continuity of a special type of linear mapping which extends the concept of algebra homomorphism: Let A and B be two independently, Koldunov [17] have shown that the inverse of a disjointness preserving bijection is disjointness preserving under certain general conditions. Disjointness preserving maps were also considered in [8] for spaces of real or complex-valued continuous functions denned on a compact Hausdorff space with the name of separating maps. The main goal in this context is to prove automatic continuity results for separating maps between different kinds of function spaces of the following type.

Abstract. We say that a linear subspace A of C 0 (X) is strongly separating if given any pair of distinct points x 1 , x 2 of the locally compact space X, then there exists f ∈ A such that |f (x 1 )| = |f(x 2 )|. In this paper we prove that a linear isometry T of A onto such a subspace B of C 0 (Y ) induces a homeomorphism h between two certain singular subspaces of the Shilov boundaries of B and A, sending the Choquet boundary of B onto the Choquet boundary of A. We also provide an example which shows that the above result is no longer true if we do not assume A to be strongly separating. Furthermore we obtain the following multiplicative representation of T : (T f)(y) = a(y)f(h(y)) for all y ∈ ∂B and all f ∈ A, where a is a unimodular scalar-valued continuous function on ∂B. These results contain and extend some others by Amir and Arbel, Holsztyński, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

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