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

@article{Peralta2012AutomaticCO, title={Automatic continuity of derivations on C*-algebras and JB*-triples}, author={Antonio M. Peralta and Bernard Russo}, journal={arXiv: Operator Algebras}, year={2012} }

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…

## 34 Citations

Local triple derivations on C*-algebras and JB*-triples

- Mathematics
- 2014

In a first result we prove that every continuous local triple derivation on a JB$^*$-triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple…

Continuity of generalized derivations on JB*-algebras

- Mathematics
- 2017

Abstract We prove that every generalized Jordan derivation D from a JB∗-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized…

Ternary Weakly Amenable C*-algebras and JB*-triples

- Mathematics
- 2012

A well known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be…

Triple derivations on von Neumann algebras

- Mathematics
- 2013

It is well known that every derivation of a von Neumann algebra into itself is an inner derivation and that every derivation of a von Neumann algebra into its predual is inner. It is less well known…

Local triple derivations from C$^*$-algebras into their iterated duals

- MathematicsTamkang Journal of Mathematics
- 2018

We show that every local triple derivation from a C$^*$-algebra into any of its iterated duals is a triple derivation. This result partially solves a problem posed by M. Burgos \emph{et al.} in…

Local Triple Derivations on Real C$$^*$$∗-Algebras and JB$$^*$$∗-Triples

- Mathematics
- 2016

We study when a local triple derivation on a real JB$$^*$$∗-triple is a triple derivation. We find an example of a (real linear) local triple derivation on a rank-one Cartan factor of type I which is…

Conjugate Linear Maps from C$$^*$$-Algebras into Their Dual Spaces which are Ternary Derivable at the Unit Element

- Mathematics
- 2020

We prove that every continuous conjugate linear mapping from a unital C
$$^*$$
-algebra A into its dual space,
$$A^*$$
, which is ternary derivable at the unit element of A is a ternary…

Derivations and Projections on Jordan Triples. An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

- Mathematics
- 2012

This paper is an elaborated version of the material presented by the author in a three hour minicourse at "V International Course of Mathematical Analysis in Andalusia," Almeria, Spain, September…

2-LOCAL TRIPLE DERIVATIONS ON

- 2014

We prove that every (not necessarily linear nor continuous) 2-local triple derivation on a von Neumann algebra M is a triple derivation, equivalently, the set Dert(M), of all triple derivations on M,…

Dual Space Valued Mappings on C $$^*$$ ∗ -Algebras Which Are Ternary Derivable at Zero

- Mathematics
- 2021

Extending derivability of a mapping from one point of a $$\mathrm C^*$$
-algebra to the entire space is one of the interesting problems in derivation theory. In this paper, by considering a…

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Ternary Weakly Amenable C*-algebras and JB*-triples

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