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

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 study the 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.