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I have the following two questions

1: Let $A$ and $B$ be Banach algebras and suppose that $B$ is semisimple. Let $T:A \to B $ be a homomorphism with $\overline {TA}=B.$ Is $T$ automatically continuous?

2: Let $A$ and $B$ be Banach algebras. Let $T:A \to B $ be a homomorphism with $\overline {TA}=B.$ Is $T$ automatically continuous?

Any help is much appreciated! Thank you!

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  • $\begingroup$ Just to clarify: is Q2 merely Q1 without the assumption that $B$ is semisimple? $\endgroup$
    – Yemon Choi
    Commented Dec 3, 2018 at 20:01
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    $\begingroup$ Have you had a look in H. Garth Dales' "Banach Algebras and Automatic Continuity"? This seems to be the reference for such questions. $\endgroup$
    – Jochen Glueck
    Commented Dec 3, 2018 at 21:39
  • $\begingroup$ @Dear yemon Choi, yes in 2 we consider B without semisimple $\endgroup$
    – user62498
    Commented Dec 4, 2018 at 6:36
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    $\begingroup$ @JochenGlueck In fairness to the OP he or she may be relying on GoogleBooks if his or her library does not have a copy of the book, and it is not one where everyone has the immediate resources to own a physical copy. ("Far too expensive" -- H. G. Dales, personal communication over coffee) $\endgroup$
    – Yemon Choi
    Commented Dec 4, 2018 at 13:53
  • $\begingroup$ To the OP: that said, if you do have access to a physical or PDF copy, please look there first $\endgroup$
    – Yemon Choi
    Commented Dec 4, 2018 at 13:53

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