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.

Caractérisations Spectrales du Radical et du Socle d’Une Paire de Jordan-Banach

RésuméIf f and g are two analytic functions from a domain D of the complex plane into respectively the Banach spaces V+ and V−, we prove that is an analytic multivalued function. From this derives the subharmonicity of the functions where ρ denotes the spectral radius. We apply these results to obtain nice caracterizations of the radical and the socle of a Banach Jordan pair, and finally we get an algebraic structural theorem.