Biderivations, local and 2-local derivation and automorphism of simple $\omega$-Lie algebras

Given a finite-dimensional complex simple $\omega$-Lie algebras $$ over $\mathbb{C}$. We prove that every local ,$2-$local derivation is a derivation and every local (resp. 2-local) automorphisms are automorphisms or an anti-automorphis (resp. automorphism). We characterize also biderivation, $\frac{1}{2}$-derivation and local (2-local) $\frac{1}{2}$-derivation of $\mathfrak{g}$.