Crossed products by twisted partial actions and graded algebras

For a twisted partial action Θ of a group G on an (associative non-necessarily unital) algebra over a commutative unital ring k, the crossed product is proved to be associative. Given a G-graded k-algebra with the mild restriction of homogeneous non-degeneracy, a criteria is established for to be isomorphic to the crossed product for some twisted partial action of G on . The equality ( g G) is one of the ingredients of the criteria, and if it holds and, moreover, has enough local units, then it is shown that is stably isomorphic to a crossed product by a twisted partial action of G.