Exemples séparant certaines classes d'algèbres topologiques

Esterle (1979) and Zelazko (1996 and 1990) gave an example of an algebra which cannot be topologized as a topological algebra (with jointly continuous multiplication), and Müller (in 1991) showed that there exists a topological algebra, which is indeed a locally bounded algebra, which cannot be topologized as a locally convex algebra.

To complete this study on the separation between the classes of algebras, we construct for every $p\in ]0,1]$ a locally bounded algebra which is $q$-normed for every $q\in ]0, \frac{p}{p+2}[$ which cannot be topologized as a locally $p$-convex algebra, and we deduce an example of a locally pseudoconvex algebra which cannot be topologized as a locally $p$-convex algebra for every $p\in ]0,1]$. We also show the existence of a topological algebra which cannot be topologized as a locally pseudoconvex algebra.