On Pre-Hilbert Noncommutative Jordan Algebras Satisfying ∥x2∥=∥x∥2

Abstract EN

Let A be a real or complex algebra.

Assuming that a vector space A is endowed with a pre-Hilbert norm ∥·∥ satisfying ∥x2∥=∥x∥2 for all x∈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,ℂ,ℍ, or ?, and A is isomorphic to ℂ in the last three cases.

These last cases permit us to show that if A is a complex pre-Hilbert noncommutative Jordan algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.

Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying ∥x2∥=∥x∥2 for all x∈A.