Zel'Manov's Theorem for Primitive Jordan-Banach Algebras

Abstract

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 the theorem are satisfied and even the inclusion A ↪→ BL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of the above theorem. 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], 1983) gave a precise description of these last algebras, it has happened only very recently that the appropriate variant of Zel’manov’s theorem for primitive Jordan algebras has been obtained (see [3] and [17]). Also very recently several particular normed versions of Zel’manov’s theorem have been provided (see [8], [6], [16], and [7]). Nevertheless, to obtain a Zel’manov type theorem for primitive Jordan-Banach algebras has remained an open problem in the last 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) which is in our opinion one of the most important novelties in the paper. Since Theorem 1 will probably have applications different from that in the paper, we have included in its statement and proof some details not strictly needed for our main purpose. The same comment should be made concerning Theorem 2, which is nothing but a fine improvement of Theorem 1 under the additional assumption of completeness. From Theorem 2 and the main results in [3], [18], and [5], the desired Jordan variant of Theorem 0 (Theorem 3) follows easily. Again roughly speaking, it asserts that primitive complex Jordan-Banach algebras, different from the simple exceptional 27-dimensional one and the simple Jordan algebras of a continuous symmetric bilinear form on a complex Banach space, can be continuously regarded as Jordan algebras of bounded linear operators ”acting irreducibly” on a suitable complex Banach space.