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

Contrary to the simple structure of tensor product quaternionic Hilbert space, octonionic situation becomes more involved. It turns out that an space can be decomposed as orthogonal direct sum two subspaces, each them isomorphic a irreducible with real space. As application, we find for given basis, Parseval equality holds if and only basis is weak associative. Fortunately, there always exists associative in This completely removes obstacles caused by failure equality.