Prime associative triple systems with nonzero socle

Von Neumann Regular Jordan Banach Triple Systems

“…By using this embedding Loos classified non-degenerate finite-dimensional ATS of the second kind over an algebraically closed field, and with other methods he studied in [16] ATS of the second kind satisfying the descending chain condition on inner ideals. Recently, the first two authors settled in [4] the structure of prime (in particular simple) ATS of the second kind with minimal inner ideals, thus extending the structure theorems of Loos already cited (see also [2] for a different approach).…”

“…In this section we develop a socle theory for non-degenerate JTS that extends that for associative and Jordan algebras [8,24,3]. The main result of this theory, the socle theorem, was suggested to the first author by McCrimmon and proved in [4] for the special case of an ATS of the second kind. We stress the description of simple components of the socle given in Proposition 2.2.…”

“…Since every simple von Neumann regular Jordan Banach triple system is reduced in the sense of [19], we devote Section 5 to classify reduced simple Jordan triple systems over an algebraically closed field. This is done by using the classification of simple Jordan triple systems given by Zelmanov in [27] and structure theorems for simple associative triple systems with minimal inner ideals (Sections 3,4). Finally, we refine this classification for the Banach case, obtaining in Section 6 that simple von Neumann regular Jordan Banach triple systems are either (i) finite-dimensional (described by Loos in [14]), (ii) Jordan triple systems coming from infinite-dimensional simple quadratic Jordan Banach algebras, or (iii) Jordan triple systems associated to infinite-dimensional simple von Neumann regular associative Banach triple systems of the second kind, which are also described.…”

Jordan-*-Triples with Minimal Inner Ideals and Compact JB*-Triples

“…We have that %!£{X, Y) under P{a)b = ab n a, and Sym{JC££ a (X), ± # ) under P(a)b = ±aba are compact Jordan-*-triples. PROPOSITION 9. Let J be a compact Jordan-*-triple.…”