Noetherian Banach Jordan pairs

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.