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On the spectrum of inner derivations in partial Jordan triples. (English) Zbl 0756.46038

Let E be a complex Banach space and E 0 a closed subspace with involution. Let (x,a,y){xa * y} be a continuous real trilinear map E×E 0 ×EE, which is symmetric complex bilinear in x, y and conjugate linear in a. Certain algebraic postulates for {xa * y} are assumed, including aa * Her(E) (aE 0 ), where aa * is the operator x{aa * x} and Her(E) is the set of all operators on E which are Hermitian in the sense of Vidav. Such systems are called here Partial J * -triples.

The main result is that when the system is geometric (all vector fields a-{xa * x}/x (aE 0 ) are complete in some bounded balanced domain in E), then every Hermitian operator aa * (aE 0 ) has a non-negative spectrum.

46L70Nonassociative selfadjoint operator algebras
46H70Nonassociative topological algebras