ON THE BEHAVIOR OF JORDAN-ALGEBRA NORMS ON ASSOCIATIVE ALGEBRAS

We prove that for a suitable associative (real or complex) algebra whi ch has many nice algebraic properties, such as being simple and having minimal idempotents, a norm can be given such that the mapping (a,b) bar arrow pointing right ab + ba is jointly continuous while (a, b) ba r arrow pointing right ab is only separately continuous. We also prove that such a pathology cannot arise for associative simple algebras wi th a unit. Similar results are obtained for the so-called ''norm exten sion problem'', and the relationship between these results and the nor med versions of Zel'manov's prime theorem for Jordan algebras are disc ussed.