Abstract
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+∑jIj) with U a closed subspace of H1 (the factor associated to the unit element in G), and any Ij a well described closed graded ideal of H, satisfying IjIk=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.
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