Para-linearity as the Nonassociative Counterpart of Linearity

Abstract

In an octonionic Hilbert space H, the octonionic inner product induces maps which fail to be octonionic linear. This fact motivates us to introduce a new notion of the octonionic para-linearity instead. To do this we encounter an insurmountable obstacle. That is, the axiom

$$\begin{aligned} \left\langle pu ,u\right\rangle =p\left\langle u ,u\right\rangle \end{aligned}$$

for any octonion p and element $u\in H$ introduced by Goldstine and Horwitz in 1964 can not be interpreted as a property to be obeyed by the octonionic para-linear maps. In this article, we solve this critical problem by showing that this axiom is in fact non-independent from others. This enables us to initiate the study of octonionic para-linear maps. We can thus establish the octonionic Riesz representation theorem which, up to isomorphism, identifies two octonionic Hilbert spaces, one of which is the dual of the other. The dual space consists of continuous left para-linear functionals and it becomes a right octonionic module under the multiplication defined in terms of the second associators which measure the failure of octonionic linearity. This right multiplication has an alternative expression

$$\begin{aligned} {(f\odot p)(x)}=pf(p^{-1}x)p, \end{aligned}$$

which is a generalized Moufang identity. Remarkably, the multiplication is compatible with the canonical norm, i.e.,

$$\begin{aligned} \left| \left| f\odot p\right| \right| =\left| \left| f\right| \right| \left| p\right| . \end{aligned}$$

Our final conclusion is that para-linearity is the nonassociative counterpart of linearity.