Infinite Dimensional Lie and Jordan Algebras

2006 ◽  
Vol 3 (2) ◽  
pp. 273-282
Author(s):  
Consuelo Martínez
Keyword(s):  
Jordan Algebras ◽  
Infinite Dimensional

Hilbert space methods in the theory of Jordan algebras. I

1975 ◽  
Vol 78 (2) ◽  
pp. 293-300 ◽  
Author(s):  
C. Viola Devapakkiam
Keyword(s):  
Hilbert Space ◽  
Classification Problem ◽  
Jordan Algebras ◽  
Inner Product ◽  
Uniqueness Problem ◽  
Associative Algebras ◽  
Infinite Dimensional

In this paper, we study the structure of certain infinite-dimensional Jordan algebras admitting an inner product. These algebras, called J*-algebras in the sequel, have already been considered in (4) in connexion with the norm uniqueness problem for non-associative algebras. We deal here with the structure and classification of these algebras. Existence of self-adjoint idempotents plays a central role in the classification problem.

scholarly journals On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mohamed Benslimane ◽  
Abdelhadi Moutassim
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Alternative Algebra ◽  
Jordan Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
First Case ◽  
Hilbert Norm ◽  
Complex Powers

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 .

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