On the structure of graded Hilbert spaces

Consider an arbitrary Hilbert space H endowed with a continuous product which induces a grading on H with respect to an abelian group G. We show that such a space H has the form H = cl ( U + ∑ j I j ) with U a closed subspace of H 1 (the factor associated to the unit element in G), and any I j a well described closed graded ideal of H , satisfying I j I k = 0 if j ≠ k . Under certain conditions, the graded simplicity of H is characterized and it is shown that H is the closure of the orthogonal direct sum of the family of its minimal (closed) graded ideals, each one being a graded simple graded Hilbert space.