a subspace of homogeneous preserving star operations

Let γ be a torsionless commutative cancellative monoid, r = lα∈γ rα be a γ-graded integral domain. in this note we show that each homogeneous star operation ⋆ : hf(r) → hf(r) of r, is the restriction of a (classical) star operation e(⋆) : f(r) → f(r) of r, that is e(⋆)|hf(r) = ⋆. we also show that the set hstarf(r) of homogeneous star operations of finite type on r, endowed with the zariski topology, is a spectral space.