Almost multiplicative functions on commutative Banach algebras

S. H. Kulkarni, and D. Sukumar. "Almost multiplicative functions on commutative Banach algebras." Studia Mathematica 197.1 (2010): 93-99. <http://eudml.org/doc/285812>.

@article{S2010,
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 < 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.},
author = {S. H. Kulkarni, D. Sukumar},
journal = {Studia Mathematica},
keywords = {commutative Banach algebras; almost commutative functionals; condition spectrum; Gleason-Kahane-Żelazko Theorem},
language = {eng},
number = {1},
pages = {93-99},
title = {Almost multiplicative functions on commutative Banach algebras},
url = {http://eudml.org/doc/285812},
volume = {197},
year = {2010},
}

TY - JOUR
AU - S. H. Kulkarni
AU - D. Sukumar
TI - Almost multiplicative functions on commutative Banach algebras
JO - Studia Mathematica
PY - 2010
VL - 197
IS - 1
SP - 93
EP - 99
AB - 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.
LA - eng
KW - commutative Banach algebras; almost commutative functionals; condition spectrum; Gleason-Kahane-Żelazko Theorem
UR - http://eudml.org/doc/285812
ER -