On a particular subspace of homogeneous preserving star operations

Let $\mathrm{\Gamma }$ be a torsionless commutative cancellative monoid, $R={⨁}_{\alpha \in \mathrm{\Gamma }}{R}_{\alpha }$ be a $\mathrm{\Gamma }$-graded integral domain. In this note we show that each homogeneous star operation $\star :\mathbf{H}\mathbf{F}(R)\to \mathbf{H}\mathbf{F}(R)$ of $R$, is the restriction of a (classical) star operation $e(\star ):F(R)\to F(R)$ of $R$. We also show that the set $HSta{r}_f(R)$ of homogeneous star operations of finite type on $R$, endowed with the Zariski topology, is a spectral space.