Strong primeness of hermitian Jordan systems
One of the main results on Jordan theory is the strongly prime structure theorem published by \textit{K. McCrimmon} and \textit{E. Zel'manov} in Adv. Math. 69, 133-222 (1988; Zbl 0656.17015). In this theorem, strongly prime Jordan algebras are classified (recalling that strongly prime = prime + nondegenerate) giving the relation of possibilities for these algebras. Most of the algebras listed there were shown to be strongly prime with a scalar extension argument. However, McCrimmon and Zel'manov remark that the only trouble is with the algebras of hermitian type, in which case this argument fails. So, their strong primeness has remained as an open problem. Similar difficulties on the description of strongly prime Jordan pairs or triple systems have been encountered. In this paper, the authors prove the strong primeness of hermitian Jordan systems (algebras, triple systems and pairs) completing Zelmanov's classification theorems. Briefly, a Jordan system of hermitian type is a Jordan system \(J\) of hermitian elements ``sandwiched'' by means of the following inclusions \(H_0(R,*)\triangleleft J\subset H(Q(R),*)\), where \(H_0(R,*)\) is an ample subspace of a \(*\)--prime associative algebra \(R\) with involution \(*\) and \(Q(R)\) is its Martindale system of symmetric quotients. In order to make the paper self-contained, a preliminary section with definitions and previous results is included. The three following sections are devoted to obtain the strongly primeness of Jordan systems, algebras, triples and pairs respectively, of hermitian type. Finally, in section four the structure theorems are reenunciated for strongly prime Jordan systems providing ``if'' as well as ``only if''. Moreover, better versions of the classifications of primitive systems are stated.
scientific article; zbMATH DE number 1106643
scientific article