Entire functions operate in complete locally A-convex algebras but not continuously. Actually squaring is not always continuous. The counterexample we give is multiplier algebra.
@article{urn:eudml:doc:41146, title = {Discontinuity of the product in multiplier algebras.}, journal = {Publicacions Matem\`atiques}, volume = {34}, year = {1990}, pages = {397-401}, zbl = {0726.46029}, language = {en}, url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41146} }
Oudadess, Mohamed. Discontinuity of the product in multiplier algebras.. Publicacions Matemàtiques, Tome 34 (1990) pp. 397-401. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41146/