Structure des algèbres de Jordan-Banach non commutatives complexes régulières ou semi-simples à spectre fini. (Structure of regular or semisimple complex noncommutative Jordan-Banach-algebras with finite spectrum)
The main result of the paper under review is the following theorem. Let A be a complex noncommutative Jordan-Banach algebra. If A is regular \((x\in U_ xA\) for all \(x\in A)\) or semisimple with finite spectrum (Spec x is finite for all \(x\in A)\), then it can be decomposed into a direct sum of a finite number of simple ideals which are either finite dimensional or finite dimensional of type \({\mathbb{C}}\oplus V\), where V is a complex Banach space endowed with a continuous nondegenerate symmetric bilinear form f and an anticommutative product \(\wedge\) such that \(f(x,x\wedge y)=0\) and \(xy=f(x,y)+x\wedge y\) for all \(x,y\in V\). This result generalizes a result of \textit{B. Aupetit} and \textit{A. Zraibi} [Manuscr. Math. 38, 381- 386 (1982; Zbl 0499.46030)].
complex noncommutative Jordan-Banach algebra
direct sum of a finite number of simple ideals
scientific article; zbMATH DE number 4132904