On Existence and Uniqueness of Universal Enveloping Locally C*-algebra for a Locally Jb-algebra

A theorem is presented on existence and uniqueness up to the topological *-isomorphism of universal locally C*-algebra for an arbitrary locally JB-algebra. The abstract Banach associative symmetrical *-algebras over C, so called C*algebras, were introduces first by Gelfand and Naimark in []. In the present time the theory of C*-algebras become a vast portion of Functional Analysis having connections and applications in almost all branches of Modern Mathematics and Theoretical Physics (see for example [] for the basic theory of C*-algebras). From the 1940’s and the beginning of 1950’s there were numerous attempts made to extend the theory of C*-algebras to a category wider than Banach algebras. For example, in 1952, while working on the theory of locally-multiplicatively-convex algebras as projective limits of projective families of Banach algebras, Arens in the paper [] and Michael in the monograph [] independently for the first time studied projective limits of projective families of functional algebras in the commutative case and projective limits of projective families of operator algebras in the noncommutative case. In 1971 Inoue in the paper [] explicitly studied topological *-algebras which are topologically *-isomorphic to projective limits of projective families of C*-algebras and obtained their basic properties. He as well suggested a name of locally C*-algebras for that category. Below we will denote these algebras as LC*-algebras. For the present state of the theory of LC*-algebras see recently published monograph of Fragoulopoulou []. At the same time there were numerous attempts to extend the theory of C*algebras to non-associative algebras which are close to associative, in particular to Jordan algebras. In fact, in 1978 Alfsen, Schultz and Størmer published their celebrated paper [], in which they introduced and studied real Jordan Banach formally real algebras called JB-algebras, which are real non-associative analogues of C*-algebras, and obtained for this category analogues of the results from aforementioned paper [] by Gelfand and Naimark. The exposition of elementary theory of JB-algebras can be found in the monograph [] by Hanche-Olsen and Størmer, published in 1984. In particular, in this monograph there is the following theorem which was for the first time proved in 1980 by Alfsen, Hanche-Olsen and Schultz in the paper []. Date: January 4, 2008. 2000 Mathematics Subject Classification. Primary 46H05, 46H70; Secondary 46L05, 46L70.