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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?

Thank you very much.

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  • $\begingroup$ It is consistent with ZF (without Axiom of Choice) that there are no discontinuous linear operators from a Banach space to a normed space. So you're not going to get any "explicit" counterexamples. $\endgroup$ – Robert Israel Feb 24 '19 at 20:36
  • $\begingroup$ Assumng ZFC, Dales has "constructed" (in the 1970s) an example of a discontinuous derivation from the disc algebra to a suitable target bimodule. The construction is supposed to be in doi.org/10.1112/plms/s3-27.4.638 but I currently don't have access to the article to check. One place you can try to find more information is the later article of Jewell: dx.doi.org/10.2140/pjm.1977.71.465 $\endgroup$ – Yemon Choi Feb 25 '19 at 1:24
  • $\begingroup$ Alternatively, it is an exercise in the book of Allan (ed. Dales) that there are discontinuous derivations from $C^n[0,1]$ into one-dimensional bimodules when $n\geq 1$: see this other MO question mathoverflow.net/questions/319859/… $\endgroup$ – Yemon Choi Feb 25 '19 at 1:26
  • $\begingroup$ Thank you Yemon Choi and Robert Israel for your valuable comments and also for giving the references. $\endgroup$ – Fermat Mar 9 '19 at 17:58
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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.

C. J. Read, Discontinuous derivations on the algebra of bounded operators on a Banach space, J. London Math. Soc. 40 (1989), 305–326.

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  • $\begingroup$ Thank you very much. @Tomek Kania $\endgroup$ – Fermat Mar 9 '19 at 18:00

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