“…In the same way as above, if ω : H → Aut(A) 0 and f : H × H → A * are two maps that satisfy the conditions (1)-(3) above, we say that (ω, f ) is a graded factor set of H in A. The corresponding generalized graded crossed product (A, H, (ω, f )) is defined also in the same way: (A, H, (ω, f )) = ⊕ σ ∈H Ax σ , where x σ are independent indeterminates on A satisfying the multiplicative conditions x σ a = ω σ (a)x σ and x σ x τ = f (σ , τ )x σ τ for all a ∈ A and σ , τ ∈ H. We recall that there is a unique graded algebra structure on (A, H, (ω, f )) which extends the grading of A and for which x σ are homogeneous elements [11,Lemma 1.2].…”