Jordan, von Neumann and Wigner [1] 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?

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

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

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