On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

Mohamed Benslimane,Abdelhadi Moutassim

doi:10.5402/2012/328752

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 .

Highlights

Let A be a real or complex algebra not necessarily associative or finite dimensional
We prove that A is finite dimensional in the following cases
Assuming that a vector space A is endowed with a pre-Hilbert norm · satisfying x2 ≤ x 2 for all x ∈ A

Summary

Introduction

Let A be a real or complex algebra not necessarily associative or finite dimensional. Zalar 1995, 1 proved that These results were extended, respectively, to the following cases. We prove that, if A is a real or complex pre-Hilbert algebra satisfying x2 ≤ x 2 for all x ∈ A, A is finite dimensional in the following cases. In the first two cases A is isomorphic to R, C, H or O and A is isomorphic to C in the last two cases This last allows us to show that if A is a complex pre-Hilbert noncommutative Jordan algebra resp., flexible algebraic algebra or Jordan algebra satisfying x2 x 2 for all x ∈ A, A is finite dimensional and is isomorphic to C Theorems 4.9 and 4.10. We give an example of an infinite-dimensional real pre-Hilbert Jordan algebra weakly alternative algebra with divisors of zero and satisfying x2 x 2 for all x ∈ A

Notation and Preliminary Results

Real Pre-Hilbert Weakly Alternative Algebras

Complex Pre-Hilbert Alternative Algebras Satisfying x2 x 2

Complexes Pre-Hilbert Powers Associative Algebras Satisfying