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Let A be a Banach algebra. A linear functional ϰ on A is called a character of A if it is multiplicative and not identical to 0 on A. This last condition is equivalent to saying that ϰ(1) = 1 because ϰ(x) = ϰ(x)ϰ(1). If ϰ is a character of A it is easy to verify that ϰ(x)∈ Sp(x), for all x ∈ A, because (x - ϰ(x)1)y = y(x - ϰ(x)1) = 1 leads to an absurdity. Consequently \(|\chi (x)| \leqslant \rho (x) \leqslant \parallel x\parallel\) so a character is continuous and of norm one.
KeywordsBanach Space Irreducible Representation Representation Theory Spectral Theory Banach Algebra
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