Almost multiplicative functions on commutative Banach algebras

Abstract

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 < e < 1. The e-condition spectrum of an element a in A is defined by σe(a) := { λ ∈ C : ‖λ− a‖ ∥∥∥(λ− a)−1∥∥∥ ≥ 1 e } . In this note, we prove following results connecting these two notions for commutative Banach algebras. (1) If φ(1) = 1 and φ is δ-almost multiplicative, then φ(a) ∈ σδ(a) for all a in A. (2) If φ is linear and φ(a) ∈ σe(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-Zelazko theorem.