Modular annihilator Jordan pairs (Q2732311)

A nondegenerate Jordan pair is said to be a modular annihilator if it is the Jacobson radical modulo its socle. This definition is compatible with that given by Barnes for associative algebras and a natural extension of that introduced by Fernández López for Jordan algebras.NEWLINENEWLINENEWLINEBarnes proved that a complex semiprimitive (associative) Banach algebra \(A\) is a modular annihilator iff it is essential, i.e., the spectrum of any element of \(A\) has a \(0\) as unique possible accumulation point. Similar characterizations of modularity were obtained by Benslimane and Rodríguez (independently, by Wilkins) for Jordan-Banach algebras, and by Hessenberger for Jordan-Banach pairs.NEWLINENEWLINENEWLINEIn the present paper, the authors study modular annihilator Jordan pairs from both algebraic and analytic point of view, the novelty of their approach being the use of local algebras of Jordan pairs.