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.

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras and Schouten algebras among others classes of algebras. The present paper is devoted to the study of the structure of Poisson color algebras of arbitrary degree, with restrictions neither on the dimension nor the base field.

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