On a particular subspace of homogeneous preserving star operations

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