Finiteness conditions in Jordan pairs
The author extends the concept of capacity from unital Jordan algebras to Jordan pairs as follows: the capacity of a Jordan pair is the infimum of the cardinalities of the finite sets of orthogonal division idempotents whose common Peirce 0-space vanishes, and the capacity is \(\infty\) if no such sets exist. The main theorem states that the capacity of a nondegenerate Jordan pair is the supremum of the lengths of all finite chains of principal inner ideals. A frame of a Jordan pair is a finite set of orthogonal local idempotents whose common Peirce 0-space lies in the radical. A strong frame is a frame of division idempotents whose common Peirce 0-space vanishes. Elementary automorphisms associated to idempotents in Jordan pairs provide the key to the two major steps in the proof of the main theorem. First, these automorphisms lead to a diagonalization procedure with respect to a strong frame in a nondegenerate Jordan pair. It follows that a nondegenerate Jordan pair of finite capacity is von Neumann regular. Second, the author uses elementary automorphisms to prove a conjugacy theorem for frames in a Jordan pair, generalizing a result of H. P. Petersson by eliminating the need to assume chain conditions.
scientific article; zbMATH DE number 4200449