Weak-local derivations and homomorphisms on C*-algebras

TL;DR

This paper proves that weak-local derivations on C*-algebras and von Neumann algebras are continuous and are actual derivations, establishing their fundamental properties and connections to automorphisms.

Contribution

It demonstrates that weak-local derivations are continuous and coincide with derivations on C*-algebras and von Neumann algebras, and explores their relation to automorphisms.

Findings

Weak-local derivations are continuous on C*-algebras.

Weak$^*$-local derivations are derivations on von Neumann algebras.

Connections between bilocal derivations and automorphisms are established.

Abstract

We prove that every weak-local derivation on a C$^*$-algebra is continuous, and the same conclusion remains valid for weak$^*$-local derivations on von Neumann algebras. We further show that weak-local derivations on C$^*$-algebras and weak$^*$-local derivations on von Neumann algebras are derivations. We also study the connections between bilocal derivations and bilocal $^*$-automorphism with our notions of extreme-strong-local derivations and automorphisms.