Examples separating certain classes of topological algebras (Q2706592)

\textit{J. Esterle} [Rev. Roum. Math. Pures Appl. 24, 1157-1164 (1979; Zbl 0447.46041)] and \textit{W. Zelazko} [Colloq. Math. 71, No. 1, 111-113 (1996; Zbl 0887.46029)] gave examples of algebras which cannot be topologized as a topological algebra (with jointly continuous multiplication) and \textit{W. Zelazko} [Proc. Am. Math. Soc. 110, No. 4, 947-949 (1990; Zbl 0727.46025)] and \textit{V. Müller} [Stud. Math. 99, No. 2, 149-153 (1991; Zbl 0764.46046)] showed that there exists a topological algebra, which cannot be topologized as a locally convex algebra. The authors continue this line of research by showing among other results, that there exist a topological algebra which cannot be topologized as a locally pseudoconvex algebra. Note that this means that the topology cannot be defined by a family of \(p_i\)-seminorms \(\|\cdot\|_i\), \(0< p_i< 1\), where as usual \(\|\cdot\|_i\) is a nonnegative subadditive map satisfying \(\|\lambda x\|_i= |\lambda|^{p_i}\|x\|_i\) for all scalar \(\lambda\).