Weakly peripherally-multiplicative mappings between uniform algebras

Weak multiplicative operators on function algebras without units

“…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. The peripheral spectrum σ π (f ) = {f (x) : |f (x)| = f }, peripherally-multiplicative operators, for which σ π (T f T g) = σ π (f g), f, g ∈ A, and weakly peripherally-multiplicative operators, for which σ π (T f T g) ∩ σ π (f g) = ∅, were introduced in [1], [8] and [7] respectively, where sufficient conditions for such operators to be algebra isomorphisms were established.…”

“…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. The peripheral spectrum σ π (f ) = {f (x) : |f (x)| = f }, peripherally-multiplicative operators, for which σ π (T f T g) = σ π (f g), f, g ∈ A, and weakly peripherally-multiplicative operators, for which σ π (T f T g) ∩ σ π (f g) = ∅, were introduced in [1], [8] and [7] respectively, where sufficient conditions for such operators to be algebra isomorphisms were established. In the case of non-unital Lipschitz algebras similar results were obtained in [6].…”

“…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

“…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.…”

Algebra isomorphisms between standard operator algebras

“…[3,4,5,8,10,13]). Rather than requiring that such a map multiplicatively preserves the entire spectrum, however, it is also natural to ask whether preserving particular subsets of the spectrum will suffice.…”