Almost multiplicative functions on commutative Banach algebras

S. H. Kulkarni ; D. Sukumar

Studia Mathematica, Tome 196 (2010), p. 93-99 / Harvested from The Polish Digital Mathematics Library

Résumé

Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map ϕ: A → ℂ is said to be δ-almost multiplicative if |ϕ(ab) - ϕ(a)ϕ(b)| ≤ δ||a|| ||b|| for all a,b ∈ A. Let 0 < ϵ < 1. The ϵ-condition spectrum of an element a in A is defined by $σ_{ϵ}(a): = {λ ∈ ℂ : ||λ-a|| ||(λ-a)^{-1}|| ≥ 1/ϵ}$ with the convention that $||λ-a|| ||(λ-a)^{-1}|| = ∞$ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If ϕ(1) = 1 and ϕ is δ-almost multiplicative, then $ϕ(a) ∈ σ_{δ}(a)$ for all a in A. (2) If ϕ is linear and $ϕ(a) ∈ σ_{ϵ}(a)$ for all a in A, then ϕ is δ-almost multiplicative for some δ. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane-Żelazko theorem.

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Publié le : 2010-01-01

Zbl 1200.46045

EUDML-ID : urn:eudml:doc:285812