On Star operation and some ideals on the Baire space
We investigate several $\sigma$-ideals on the Baire space $\omega^\omega$ $(\mathbb{Z}^\omega)$, introducing and studying the ideals $\mathcal{G}$ and $\mathcal{SMZ}^+$, alongside the classical ideals of meager sets, strong measure zero sets, the eventually different ideal and infinitely equal ideal We establish structural relationships and proper inclusions among these ideals. Also we compute the cardinal invariants of $\mathcal{M}_-$, proving that they are same as invariants of $\sigma$-ideal of meager sets. We further analyze the operation $^*$ on families of sets, establishing dual relationships such as $\mathcal{ED}^* = \mathcal{IE}$, $\mathcal{IE}^*=\mathcal{ED}$, $\mathcal{M}_-^*=\mathcal{SMZ}^+$ and $\mathcal{H}^* = \mathcal{G}$, and derive separations between ideals under additional set-theoretic assumptions. Finally, we prove a tree dichotomy theorem for the ideal $\mathcal{M}_-$ and we study the associated forcing notion.