Automatic continuity of certain isomorphisms between regular Banach function algebras

Abstract

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.

2-Local automorphisms of operator algebras

“…Indeed, a little more can be said with the same argument. One can find a generalized version of the Banach-Stone theorem in [4] to finish a proof of the following Proposition 4.1. Every surjective 2-local automorphism of a regular uniform real or complex function algebra is an automorphism.…”

Disjointness preserving mappings on BSE Ditkin algebras

“…In [9], we extended the definition of disjointness preserving mappings to the class of regular Banach function algebras. Let us recall that a linear map T defined from a regular Banach function algebra A into such an algebra B is said to be disjointness preserving or separating if f · g ≡ 0 implies T (f ) · T (g) ≡ 0 for all f, g ∈ A.…”

“…Next we gather the main results concerning disjointness preserving maps between regular Banach function algebras, which can be found in [9]:…”

Completely bounded disjointness preserving operators between Fourier algebras

“…It is clear that a homomorphism preserves disjointness of cozero sets of functions, hence is a disjointness preserving operator. In [6,7], Font studied bounded disjointness preserving bijections between Fourier algebras, and showed that such an operator is a weighted composition operator. In a recent paper [1], the authors introduced the concept of property (A) and studied such maps in the setting of Banach algebras with property (A).…”

“…In this paper, we completely characterize such operators in the case the involved groups are amenable. Our characterization takes into account the algebraic structure of the underlying groups, and not only their topology, which was the primary emphasis in [6,7].…”