We construct a complete multiplicatively pseudoconvex algebra with the property announced in the title. This solves Problem 25 of [6].
@article{bwmeta1.element.bwnjournal-article-smv120i1p89bwm,
author = {W. \.Zelazko},
title = {A non-locally convex topological algebra with all commutative subalgebras locally convex},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {89-94},
zbl = {0881.46035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv120i1p89bwm}
}
Żelazko, W. A non-locally convex topological algebra with all commutative subalgebras locally convex. Studia Mathematica, Tome 119 (1996) pp. 89-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv120i1p89bwm/
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