# Formally real non-Jordan algebras

Jordan, von Neumann and Wigner  showed that for any finite-dimensional real vector space $$A$$ with a bilinear commutative power-associative operation $$\circ : A \times A \to A$$, the formal reality condition

$$a_1 \circ a_1 + \cdots + a_n \circ a_n = 0 \implies a_1, \dots, a_n = 0$$

implies the remaining Jordan algebra axiom: for all $$a \in A$$, multiplication by $$a \circ a$$ commutes with multiplication by $$a$$.

Thus, in the definition of a formally real Jordan algebra (which is the kind of Jordan algebra important in quantum mechanics), we can leave out the law $$(a \circ a) \circ (a \circ b) = a \circ ((a \circ a) \circ b)$$ without harm, since it follows from the other conditions... in the finite-dimensional case.

But their proof uses finite-dimensionality. So, my question is:

Do you know an example of an infinite-dimensional vector space with a commutative bilinear power-associative operation obeying the formal reality condition that is not a Jordan algebra?

 P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29-64.

• I'd try the free commutative power-associative $\mathbf{R}$-algebra on 2 generators. – YCor Jun 8 '20 at 22:52