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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 ${\sigma }_{ϵ}\left(a\right):=\lambda \in ℂ:\left|\right|\lambda -{a\left|\right|\left|\right|(\lambda -a)}^{-1}\left|\right|\ge 1/ϵ$ with the convention that $\left|\right|\lambda -{a\left|\right|\left|\right|(\lambda -a)}^{-1}\left|\right|=\infty$ 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 $\varphi \left(a\right)\in {\sigma }_{\delta }\left(a\right)$ for all a in A. (2) If ϕ is linear and $\varphi \left(a\right)\in {\sigma }_{ϵ}\left(a\right)$ for all a in A,...