On infinite dimensional algebras with regular gradings

Let G be a finite abelian group and let K be an algebraically closed field of characteristic 0. We consider associative unital algebras A over K graded by G , that is A=⊕ g∈ G Ag A = ⊕ g ∈ G A g , where the vector subspaces Ag A g satisfy AgAₕ⊆ Ag+h A g A h ⊆ A g + h for every g , h∈ G h ∈ G . Such a G -grading is called regular whenever for every n -tuple (g₁,… ,gₙ)∈ Gⁿ ( g 1 , … , g n ) ∈ G n there exist homogeneous elements aᵢ∈ Agᵢ a i ∈ A g i such that a₁⋯ aₙ≠ 0 a 1 ⋯ a n ≠ 0 in A ; furthermore, for every g , h∈ G h ∈ G and every ag∈ Ag a g ∈ A g , aₕ∈ Aₕ a h ∈ A h