Almost multiplicative functions on commutative Banach algebras

• Autorzy: S. H. Kulkarni , D. Sukumar
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46J05: General theory of commutative topological algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 197
• Numer: 1
• Strony: 93-99

Daty

wydano

2010

Twórcy

autor

S. H. Kulkarni

• Indian Institute of Technology Madras, Chennai, India

autor

D. Sukumar

• National Institute of Technology Karnataka, Surathkal, India

Identyfikatory

DOI

10.4064/sm197-1-8