Strongly graded Poisson-Hilbert spaces

Abstract

Let H be an arbitrary Hilbert space endowed with an associative continuous product which induces a grading on H with respect to an abelian group G. If H is also endowed with a continuous Lie product compatible with the associative product, via a Leibniz identity, and with the grading in a strong way we are dealing with a strongly graded Poisson-Hilbert space. We focuss on the structure of this category of spaces by showing that if H is a strongly graded Poisson-Hilbert space with zero annihilator, of maximal length and with a coherent 1-homogeneous space, then H can be written as the orthogonal direct sum of closed ideals, each one being a simple strongly graded Poisson-Hilbert space.