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Almost multiplicative functions on commutative Banach algebras

S. H. KulkarniD. Sukumar — 2010

Studia Mathematica

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 : | | 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,...

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