A non-Banach in-convex algebra all of whose closed commutative subalgebras are Banach algebras.
Studia Mathematica, Tome 119 (1996) no. 2, p. 195.

Voir la notice de l'article dans European Digital Mathematics Library

We construct two examples of complete multiplicatively convex algebras with the property that all their maximal commutative subalgebras and consequently all commutative closed subalgebras are Banach algebras. One of them is non-metrizable and the other is metrizable and non-Banach. This solves Problems 12-16 and 22-24 of [7].
Classification : 46H05, 46H10, 46J30
Mots-clés : topological algebra, -convex, complete multiplicatively convex algebras, maximal commutative subalgebras, non-metrizable, non-Banach 
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     author = {W. \.Zelazko},
     title = {A {non-Banach} in-convex algebra all of whose closed commutative subalgebras are {Banach} algebras.},
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     publisher = {mathdoc},
     volume = {119},
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     year = {1996},
     zbl = {0879.46023},
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W. Żelazko. A non-Banach in-convex algebra all of whose closed commutative subalgebras are Banach algebras.. Studia Mathematica, Tome 119 (1996) no. 2, p. 195. https://geodesic-test.mathdoc.fr/item/STUMA_1996__119_2_216294/