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Distinguishing Jordan polynomials by means of a single Jordan-algebra norm. (English) Zbl 0887.46031

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.

MSC:

46H70 Nonassociative topological algebras
17C65 Jordan structures on Banach spaces and algebras

Citations:

Zbl 0852.17033
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