<abstract><p>In this study, a $ \zeta^*_\Gamma $-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 $ \Psi_{\zeta^*_\Gamma} $ operator is defined and topologies are obtained. Moreover, alternative answers are given to an open question found in the literature. $ \Psi_{\zeta^*_\Gamma} $-compatibility is defined and its properties are examined. $ \Psi_{\zeta^*_\Gamma} $-compatibility is characterized with the help of the new operator. Finally, new spaces were defined and characterized.</p></abstract>