International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 15, Pages 2447-2460
doi:10.1155/IJMMS.2005.2447
Gleason-kahane-Żelazko theorem for spectrally bounded algebra
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
Received 29 December 2004; Revised 15 June 2005
Copyright © 2005 S. H. Kulkarni and D. Sukumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.