Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^\*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous theorem due to Johnson and Sinclair states that every derivation on a semisimple Banach algebra is continuous. Generalizing theses two resutls, since any $C^\*$-algebra is semisimple, one asks if all derivations from semisimple Banach algebras into their Banach bimodules are continuous. If the answer to this question is negative, is it always possible to construct such a discontinuous derivation? 


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Continuity of the derivations from semisimple Banach algebras
Charles Read constructed a Banach space $E$ such that the algebra $\mathcal{B}(E)$ of all bounded linear operators on $E$ admits a discontinuous derivation. Certainly, $\mathcal{B}(E)$ is Jacobson-semisimple. Note that Read's space is quite exotic as this phenomenon does not occur on classical Banach spaces such as $L\_p$-spaces or $C(K)$-spaces for $K$ compact metric.



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> C. J. Read, [Discontinuous derivations on the algebra of bounded operators on a Banach space](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-40.2.305), *J. London Math. Soc.* **40** (1989), 305–326.
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