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ISRN Algebra
Volume 2012 (2012), Article ID 328752, 11 pages
http://dx.doi.org/10.5402/2012/328752
Research Article

On Pre-Hilbert Noncommutative Jordan Algebras Satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2

Département de Mathématiques et Informatique, Faculté des Sciences, B.P. 2121, Tétouan, Morocco

Received 17 April 2012; Accepted 30 May 2012

Academic Editors: A. Jaballah, A. Kiliçman, D. Sage, K. P. Shum, F. Uhlig, A. Vourdas, and H. You

Copyright © 2012 Mohamed Benslimane and Abdelhadi Moutassim. 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

Let 𝐴 be a real or complex algebra. Assuming that a vector space 𝐴 is endowed with a pre-Hilbert norm β€– β‹… β€– satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . We prove that 𝐴 is finite dimensional in the following cases. (1) 𝐴 is a real weakly alternative algebra without divisors of zero. (2) 𝐴 is a complex powers associative algebra. (3) 𝐴 is a complex flexible algebraic algebra. (4) 𝐴 is a complex Jordan algebra. In the first case 𝐴 is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 , and 𝐴 is isomorphic to β„‚ in the last three cases. These last cases permit us to show that if 𝐴 is a complex pre-Hilbert noncommutative Jordan algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 .