Jacobson's theorem on derivations of primitive rings with nonzero socle: a proof and applications
Identificadores
URI: https://hdl.handle.net/10481/84432Metadatos
Mostrar el registro completo del ítemEditorial
Springer
Materia
Primitive ring Differential operator Additive derivation Standard operator ring Quaternionic normed space H*-algebra
Fecha
2023-07-28Referencia bibliográfica
European Journal of Mathematic (2023) 9:67
Patrocinador
Junta de Andalucía Grant FQM199; Universidad de Granada/CBUAResumen
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 so-called 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\to AB-BA for
some continuous linear operator B on the space. Our quaternionic approach
allows us to generalize Rickart'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 one-to-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.