Multiplicatively range-preserving maps between Banach function algebras

Abstract

Let A and B be two Banach function algebras on locally compact Hausdorff spaces X and Y , respectively. Let T be a multiplicatively range-preserving map from A onto B in the senseWe define equivalence relations on appropriate subsets X and Y of X and Y , respectively, and show that T induces a homeomorphism between the quotient spaces of X and Y by these equivalence relations. In particular, if all points in the Choquet boundaries of A and B are strong boundary points, then X and Y are equal to the Choquet boundaries of A and B, respectively, and moreover, there exist a continuous function h on the Choquet boundary of B taking its values in {−1, 1} and a homeomorphism ϕ from the Choquet boundary of B onto the Choquet boundary of A such that T f (y) = h( y) f (ϕ(y)) for all f ∈ A and y in the Choquet boundary of B. For certain Banach function algebras A and B on compact Hausdorff spaces X and Y , respectively, we can weaken the surjectivity assumption and give a representation for maps belonging 2-locally to the family of all multiplicatively range-preserving maps from A onto B.

Weak multiplicative operators on function algebras without units

“…If h ∈ F ψ(y) (A) then k = Φ(|h|) ∈ |F y (B)| by (4). Therefore, |h(ψ(y))| = 1 = (Φ(|h|))(y), and hence (Φ(|h|))(y) = |h(ψ(y))|…”

“…In [11] Rao and Roy extended this result for a self-map of a uniform algebra, and in [12] for a self-map of a function algebra without unit. In [2] it was proven for surjections between distinct uniform algebras, in [3] for surjections between semisimple commutative Banach algebras with units, and in [4] between completely regular commutative Banach algebras without units. Norm-multiplicative operators, for which T f T g = f g , f, g ∈ A, were introduced in [7], where sufficient conditions for a norm-multiplicative operator between uniform algebras to be a composition operator in modulus were obtained.…”

“…For the proof we will use, with necessary adjustments, the technique developed in [11] and widely used later in [2,4,6,8,7,12]. First we establish several preliminary lemmas, in all of which A, B and Φ are as in Proposition 2.4.…”

Norm-additive in modulus maps between function algebras

“…Using a similar technique to [10], we introduce appropriate subsets X and Y of X and Y , respectively, with some equivalence relations on them (we include all statements and details for the sake of completeness) and show that surjective jointly norm-additive in modulus maps T, S : A −→ B induce a homeomorphism between the produced quotient spaces. In particular, if all points in Ch(A) and Ch(B) are strong boundary points, then X = Ch(A), Y = Ch(B) and each equivalence class consists of just one element.…”

“…Motivated by Molnár's result, some extensions to the context of uniform algebras and Banach function algebras have been given in [8,9,10,23,24] and in [18,19] with respect to a part of the spectrum or the range.…”