Uniform algebra isomorphisms and peripheral multiplicativity

Abstract

Abstract. Let ϕ : A → B be a surjective operator between two uniform algebras with ϕ(1) = 1. We show that if ϕ satisfies the peripheral multiplicativity conditionwhere σ π (f ) is the peripheral spectrum of f , then ϕ is an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.An important question in Banach algebra theory, which still lacks a satisfactory answer, is to find criteria for an operator between two Banach algebras to be linear and multiplicative. For linear operators between semisimple algebras an answer is suggested by the theorem of Gleason-Kahane-Żelazko (e.g., [11]) in terms of spectra of algebra elements. A theorem by Kowalski and S lodkowski [6] considers alternative spectral conditions for not necessarily linear operators. N. V. Rao and A. K. Roy [9] have introduced an interesting spectral multiplicativity condition that contributes to the matter. In particular for unital, that is Φ(1) = 1, operators Φ they have proven the following:If A is a uniform algebra on X and Φ : A → A is a surjective unital operator such that where Ran (f ) = f (X) is the range of f . N. V. Rao, T. Tonev and E. Toneva [10] (see also [2]) considered a spectral additivity condition related to the peripheral

Norm-additive in modulus maps between function algebras

“…It should be noted that the concept of the peripheral range of a function was introduced by Luttman and Tonev in [19].…”

“…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

“…For maps presumed from the beginning to be linear it has been an active area of research, particularly for maps that also preserve some non-algebraic structure, the so-called linear preservers (see e.g. [9,10,12]). Recall that the spectrum of an operator A ∈ B(X) is the compact set σ(A) = {λ ∈ C : (λI −A) / ∈ B(X) −1 }.…”

“…[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.…”

“…The peripheral spectrum σ π (A) = {λ ∈ σ(A) : |λ| = max z∈σ(A) |z|} of A is the set of spectral values of A of maximum modulus. In [3,10] Molnár's result is extended to surjective maps ϕ : A → B between two uniform algebras that instead of (1) satisfy the equation…”