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 A over a commutative unital ring k, the crossed product A⋊ΘG is proved to be associative. Given a G-graded k-algebra B=⊕g∈GBg with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B1⋊ΘG for some twisted partial action of G on B1. The equality BgBg−1Bg=Bg (∀g∈G) is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.