Quasi-linear functionals determined by weak-2-local $^*$-derivations on $B(H)$

TL;DR

This paper proves that weak-2-local $^*$-derivations on $B(H)$ are linear $^*$-derivations for separable Hilbert spaces and that weak-2-local derivations on finite-dimensional C$^*$-algebras are linear derivations, clarifying their structure.

Contribution

It establishes the linearity of weak-2-local $^*$-derivations on $B(H)$ and finite-dimensional C$^*$-algebras, extending understanding of their algebraic properties.

Findings

Weak-2-local $^*$-derivations on $B(H)$ are linear $^*$-derivations.

Weak-2-local derivations on finite-dimensional C$^*$-algebras are linear derivations.

The results unify the behavior of weak-2-local derivations across different algebra classes.

Abstract

We prove that, for every separable complex Hilbert space $H$, every weak-2-local $^*$-derivation on $B(H)$ is a linear $^*$-derivation. We also establish that every (non-necessarily linear nor continuous) weak-2-local derivation on a finite dimensional C$^*$-algebra is a linear derivation.