Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras

TL;DR

This paper explores the structure of octonionic Hilbert spaces, revealing their decomposition, conditions for Parseval equality, and applications to Cayley-Dickson algebras, including basis construction and algebra analysis.

Contribution

It introduces a novel decomposition of octonionic Hilbert spaces, characterizes Parseval equality conditions, and applies these findings to Cayley-Dickson algebras with explicit basis construction.

Findings

Octonionic Hilbert space decomposes into two isomorphic subspaces.

Parseval equality holds if and only if the basis is weak associative.

Explicit weak associative orthonormal basis constructed in Cayley-Dickson algebras.

Abstract

Contrary to the simple structure of the tensor product of the quaternionic Hilbert space, the octonionic situation becomes more involved. It turns out that an octonionic Hilbert space can be decomposed as an orthogonal direct sum of two subspaces, each of them isomorphic to a tensor product of an irreducible octonionic Hilbert space with a real Hilbert space. As an application, we find that for a given orthogonal basis the octonionic Parseval equality holds if and only if the basis is weak associative. Fortunately, there always exists a weak associative orthogonal basis in an octonionic Hilbert space. This completely removes the obstacles caused by the failure of the octonionic Parseval equality. As another application, we provide a new approach to studythe Cayley-Dickson algebras, which turn out to be specific examples of octonionic Hilbert spaces. An explicit weak associative orthonormal basis is constructed in each Cayley-Dickson algebra.