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Gleason-Kahane-Żelazko theorem for spectrally bounded algebra
, D. Sukumar
Published in
2005
Volume: 2005
   
Issue: 15
Pages: 2447 - 2460
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 ab = ba and a2 + b2 is invertible. Then φ(ab) = 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. Copyright © 2005 Hindawi Publishing Corporation.
About the journal
JournalInternational Journal of Mathematics and Mathematical Sciences
ISSN01611712