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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