Zel’manov’s Theorem for Primitive Jordan–banach Algebras

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A9BL(X ) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0. The notion of primitiveness for Jordan algebras was introduced and developed in 1981 by L. Hogben and K. McCrimmon [10]. Primitive Jordan algebras are relevant particular types of prime nondegenerate Jordan algebras, but, although the celebrated Zel’manov prime theorem [19] gave a precise description of these last algebras in 1983, the appropriate variant of Zel’manov’s theorem for primitive Jordan algebras has been obtained only very recently (see [1, 17]). Also very recently, several particular normed versions of Zel’manov’s theorem have been provided (see [5, 6, 8, 16]). Nevertheless, a Zel’manov type theorem for primitive Jordan–Banach algebras has remained an open problem in the last few years [15]. In fact, we have been able to prove such a theorem, but only passing through a general normed version of the Zel’manov prime theorem (see Theorem 1 below), which is in our opinion one of the most important novelties in this paper. Since Theorem 1 will probably have applications that differ from the one in this paper, we have included in its statement and proof some details that are not strictly needed for our main purpose. The same comment applies to Theorem 2 below, which is nothing but a fine improvement of Theorem 1 under the additional assumption of completeness. From Theorem 2 and the main results in [1, 4, 18], the desired Jordan variant of Theorem 0 (Theorem 3 below) follows easily. Roughly speaking, this asserts that primitive complex Jordan–Banach algebras, which are different from