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We prove by elementary methods the following generalization of a theorem due to Glea-son, Kahane, and ˙ Zelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ : A → C be a linear map such that φ(1) = 1 and (φ(a)) 2 + (φ(b)) 2 = 0 for all a, b in A satisfying ab = ba and a 2 + b 2 is invertible. Then… (More)

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