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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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  • Just to clarify: is Q2 merely Q1 without the assumption that $B$ is semisimple? – Yemon Choi Dec 03 '18 at 20:01
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    Have you had a look in H. Garth Dales' "Banach Algebras and Automatic Continuity"? This seems to be the reference for such questions. – Jochen Glueck Dec 03 '18 at 21:39
  • @Dear yemon Choi, yes in 2 we consider B without semisimple – user62498 Dec 04 '18 at 06:36
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    @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) – Yemon Choi Dec 04 '18 at 13:53
  • To the OP: that said, if you do have access to a physical or PDF copy, please look there first – Yemon Choi Dec 04 '18 at 13:53

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