On Pre-Hilbert Noncommutative Jordan Algebras Satisfying ∥x2∥=∥x∥2. (22nd August 2012) - Benslimane, Mohamed; Moutassim, Abdelhadi
Abstract : LetA be a real or complex algebra. Assuming that a vector spaceA is endowed with a pre-Hilbert norm∥ · ∥ satisfying∥ x 2 ∥ = ∥ x ∥ 2 for allx ∈ A . We prove thatA 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 caseA is isomorphic toℝ, ℂ, ℍ, or𝕆, andA is isomorphic toℂ in the last three cases. These last cases permit us to show that ifA is a complex pre-Hilbert noncommutative Jordan algebra satisfying∥ x 2 ∥ = ∥ x ∥ 2 for allx ∈ A, thenA is finite dimensional and is isomorphic toℂ . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying∥ x 2 ∥ = ∥ x ∥ 2 for allx ∈ A .