Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications
Metadatos
Afficher la notice complèteEditorial
Springer Nature
Materia
Primitive ring Differential operator Additive derivation Standard operator ring H∗-algebra
Date
2023-07-28Referencia bibliográfica
Rodríguez Palacios, Á., Cabrera García, M. Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications. European Journal of Mathematics 9, 67 (2023). [https://doi.org/10.1007/s40879-023-00667-4]
Patrocinador
Universidad de Granada/CBUARésumé
We provide a proof of Jacobson’s theorem on derivations of primitive rings with
nonzero socle. Both Jacobson’s theorem and its formulation (in terms of the socalled
differential operators on left vector spaces over a division ring) underlie our
paper. We apply Jacobson’s theorem to describe derivations of standard operator
rings on real, complex, or quaternionic left normed spaces. Indeed, when the space is
infinite-dimensional, every derivation of such a standard operator ring is of the form
A → AB− BA for some continuous linear operator B on the space. Our quaternionic
approach allows us to generalizeRickart’s theorem on representation of primitive complete
normed associative complex algebras with nonzero socle to the case of primitive
real or complex associative normed Q-algebras with nonzero socle. We prove that
additive derivations of the Jordan algebra of a continuous nondegenerate symmetric
bilinear form on any infinite-dimensional real or complex Banach space are in a oneto-
one natural correspondence with those continuous linear operators on the space
which are skew-adjoint relative to the form. Finally we prove that additive derivations
of a real or complex (possibly non-associative) H∗-algebra with no nonzero
finite-dimensional direct summand are linear and continuous