Noetherian Banach Jordan pairs

Abstract

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.Here we show the following results:(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

The Jordan algebras of a Lie algebra

“…Local algebras (or their related notion of subquotient) have also proved their usefulness in some questions involving Jordan Banach systems. For instance, in the solution given by Loos to the problem on the coincidence of the socle with the largest properly spectrum-finite ideal of a semiprimitive Banach Jordan pair [21] (see also [14]); in the proof of a structure theorem for Noetherian Banach Jordan pairs [8]; and in the solution to the problem on automatic continuity of derivations on semiprimitive Banach Jordan pairs [15]. On the other hand, ad-nilpotent elements of index at most 3 (here called Jordan elements) play a fundamental role in the proof of Kostrikin's conjecture that any finite-dimensional simple nondegenerate Lie algebra (over a field of characteristic greater than 5) is classical [6,28].…”