Weak 2-Local Inner Derivations of Semiprime ∗-Banach Algebras

This paper introduces and examines the concept of weak 2-local inner derivations and their relationship to P-2-local mappings. It is established that every such mapping on a semiprime ∗-Banach algebra with a faithful trace is a derivation, which also provides a complete characterization on finite von Neumann algebras. Additionally, it is shown that weak 2-local inner derivations coincide with the 2-local reflection closure of the inner derivations.