It is well known that characters on real or complex Banach algebra are automaticaly continuous. In His1959 doctoral dissertation, Ernset Michael conjectured that the same result on automatic continuity of characters on a complex Fréchet algebra would remain true.
In his very recent talk in Seoul
https://sites.google.com/site/2019ilju/
Richard Aron mentioned this conjecture as an open problem.
It seems that the problem is still open. Some attempt to solve it had been made some years ago, but it seems that there were a gap in the proof.
However, the problem has been shown to be equivalent to the same problem in a special algebra by J. Esterle and others. See
https://projecteuclid.org/download/pdf_1/euclid.bams/1183553468
https://projecteuclid.org/download/pdf_1/euclid.afa/1399899933
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