Automatic continuity of derivations on C*-algebras and JB*-triples

TL;DR

This paper establishes conditions for the automatic continuity of derivations on JB*-triples and C*-algebras, showing that many derivations are inherently continuous, thus resolving longstanding open problems in the field.

Contribution

It introduces Jordan triple modules and proves that derivations into duals are automatically continuous, solving a ten-year-old open problem.

Findings

Derivations from JB*-triples into dual spaces are continuous.

Triple derivations from C*-algebras into Banach modules are continuous.

Jordan derivations from C*-algebras are actual derivations.

Abstract

We introduce the notion of a Jordan triple module and determine the precise conditions under which every derivation from a JB*-triple E into a Banach (Jordan) triple E-module is continuous. In particular, every derivation from a real or complex JB*-triple into its dual space is automatically continuous. Among the consequences, we prove that every triple derivation from a C*-algebra A to a Banach triple A-module is continuous. In particular, every Jordan derivation from A to a Banach A-bimodule is a derivation, a result which complements a classical theorem due to B.E. Johnson and solves a problem which has remained open for over ten years.