Abstract
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.
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© 1991 Springer-Verlag New York, Inc.
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Aupetit, B. (1991). Representation Theory. In: A Primer on Spectral Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3048-9_4
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DOI: https://doi.org/10.1007/978-1-4612-3048-9_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97390-6
Online ISBN: 978-1-4612-3048-9
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