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ISRN Algebra
Volume 2012 (2012), Article ID 328752, 11 pages
doi:10.5402/2012/328752
On Pre-Hilbert Noncommutative Jordan Algebras Satisfying
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 be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .