In this study, a ${\zeta }_{\mathrm{\Gamma }}^{\ast }$-local function is defined and its properties are examined. This newly defined local function is compared with the well-known local function and the local closure function according to the relation of being a subset. With the help of this new local function, the ${\mathrm{\Psi }}_{{\zeta }_{\mathrm{\Gamma }}^{\ast }}$ operator is defined and topologies are obtained. Moreover, alternative answers are given to an open question found in the literature. ${\mathrm{\Psi }}_{{\zeta }_{\mathrm{\Gamma }}^{\ast }}$-compatibility is defined and its properties are examined. ${\mathrm{\Psi }}_{{\zeta }_{\mathrm{\Gamma }}^{\ast }}$-compatibility is characterized with the help of the new operator. Finally, new spaces were defined and characterized.