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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