Operator Commutativity in Jordan algebras

Abstract

If a and b are elements of a Jordan algebra $\mathfrak{A}$ we say that a and b operator-commute or o-commute if the multiplications R _a and R _b commute. Here R _a is the linear transformation x→xa = ax of $\mathfrak{A}$. The notion of o-commutativity has been introduced by Jordan, Wigner, and von Neumann [4] who called this concept simply commutativity. Since every Jordan algebra is commutative in the usual sense, the above change in terminology seems to be desirable. In this note we shall study the notion of o-commutativity for finite-dimensional Jordan algebras of characteristic 0. Our results are based on those of two previous papers [l; 2].