Weak-local derivations and homomorphisms on C*-algebras

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.

Keywords:

• weak-local derivation
• weak *-local derivation
• weak-local *-automorphism
• weak *-local *-automorphism
• extreme-weak*-local derivation
• extreme-weak*-local *-automorphism

AMS Subject Classifications:

• Primary: 47B47
• 46L57
• 47B44
• 47B48
• 47B49
• Secondary: 15A86
• 47L10

View correction statement:

Weak-local derivations and homomorphisms on C*-algebras

Acknowledgements

We thank the referee for providing constructive comments and help in improving the contents of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Authors partially supported by the Spanish Ministry of Science and Innovation, D.G.I. project no. MTM2011-23843, Junta de Andalucía [grant number FQM375]. Second author partially supported by the Deanship of Scientific Research at King Saud University (Saudi Arabia) research group no. RG-1435-020. The first author acknowledges the partial financial support from the IEMath-GR program for visits of young talented researchers, and from the Higher Education And Scientific Research In Tunisia project UR11ES52: Analyse, Géométrie et Applications.