Orthogonal-gradings on $$H^*$$H∗-algebras

Abstract

We study set-gradings on proper $$H^*$$ -algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary $$H^*$$ -algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary $$H^*$$ -algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative $$H^*$$ -algebra, or a $${\mathbb Z}_2^3$$ -grading of the complex octonions $${\mathbb O}$$ or a non-group grading which is a refinement of the Peirce decomposition of $${\mathbb O}$$ respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.