We prove that, if A is an associative algebra with two commuting involutions τ and π, if A is a τ-π-tight envelope of the Jordan Triple System T:=H(A,τ) ∩ S(A,π), and if T is nondegenerate, then every complete norm on T making the triple product continuous is equivalent to the restriction to T of an algebra norm on A.

For = ℝ or ℂ we exhibit a Jordan-algebra norm ⎮·⎮ on the simple associative algebra ${M}_{\infty}\left(\right)$ with the property that Jordan polynomials over are precisely those associative polynomials over which act ⎮·⎮-continuously on ${M}_{\infty}\left(\right)$. This analytic determination of Jordan polynomials improves the one recently obtained in [5].

We prove that there exists a real or complex central simple associative algebra M with minimal one-sided ideals such that, for every non-Jordan associative polynomial p, a Jordan-algebra norm can be given on M in such a way that the action of p on M becomes discontinuous.

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