A Vidav–Palmer Theorem for Jordan C∗‐Algebras and Related Topics

A result of J. Martinez, A. Mojtar and the author (see [10]) proves that the natural involution of a nonassociative V-algebra A is multiplicative if and only if A is a noncommutative Jordan algebra (thus determining the widest context in which assertion (a) is valid in the nonassociative case). So noncommutative Jordan Valgebras are a good generalization of associative C*-algebras (note in addition their good algebraic behaviour: power-associativity, existence of an available concept of inverse for some of their elements, etc.). The purpose of this paper is to continue the study of noncommutative Jordan V-algebras begun in [10]. The basic tool is the recent theorem of the Vidav-Palmer type due to J. Martinez [8] and M. A. Youngson ([19] and [20]) showing that the class of (commutative) Jordan V-algebras is just the one of Jordan C*-algebras. The concept of Jordan C*algebra has been formulated by Kaplansky using axioms which we can call "of the Gelfand-Naimark type" and it has been studied by several authors since the work of Wright [15] whose main result establishes that real JB-algebras in the sense of Alfsen, Schultz and Stormer [1] are just the selfadjoint parts of Jordan C*-algebras. In Section 1 of this paper I give a new proof of the above-mentioned result by Martinez and Youngson (Theorem 3) and from this and the papers [10], [15] and [17] we obtain interesting consequences in the theory of V-algebras, especially of noncommutative Jordan V-algebras (Theorems 8 and 10) which are useful in the following sections.