Moreno Galindo, A. Distinguishing Jordan polynomials by means of a single Jordan-algebra norm. (English) Zbl 0887.46031 Stud. Math. 122, No. 1, 67-73 (1997). The question about the continuity of the action of particular non-Jordan associative polynomials on suitable associative algebras endowed with Jordan-algebra norms is related to the search for normed versions of the Zelmanov prime theorem for Jordan algebras.Let \(M_\infty\) be the simple associative algebra of all countably infinite matrices over the real or complex field with a finite number of nonzero entries. M. Cabrera García, A. Moreno Galindo, A. Rodríguez Palacios and E. I. Zel’manov [Stud. Math. 117, No. 2, 137-147 (1996; Zbl 0852.17033)] proved that, for any non-Jordan associative polynomial \(p\), a Jordan-algebra norm can be defined on \(M_\infty\) such that the action of \(p\) on the selfadjoint part of \(M_\infty\) becomes discontinuous. Refining this idea, the author constructs in this paper a Jordan-algebra norm on \(M_\infty\) making the action of any non-Jordan associative polynomial on it discontinuous. Moreover, he shows that there exists a Jordan subalgebra of \(M_\infty\) which is not invariant under any non-Jordan associative polynomial, and that, for a suitable associative norm on \(M_\infty\), it is closed. Reviewer: A.Fernández López (Malaga) MSC: 46H70 Nonassociative topological algebras 17C65 Jordan structures on Banach spaces and algebras Keywords:non-Jordan associative polynomials; associative algebras; Jordan-algebra norms; Zelmanov prime theorem; Jordan algebras Citations:Zbl 0852.17033 PDFBibTeX XMLCite \textit{A. Moreno Galindo}, Stud. Math. 122, No. 1, 67--73 (1997; Zbl 0887.46031) Full Text: DOI EuDML