Gleason-kahane-Żelazko theorem for spectrally bounded algebra

S. H. Kulkarni and D. Sukumar

International Journal of Mathematics and Mathematical Sciences, 2005, vol. 2005, 1-14

Abstract:

We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. 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 a b = b a and a 2 + b 2 is invertible. Then φ ( a b ) = φ ( a ) φ ( b ) for all a , b in A . Similar results are proved for real and complex algebras using Ransford's concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum.

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:985304

DOI: 10.1155/IJMMS.2005.2447

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