A generalization of a theorem of Chernoff on standard operator algebras

Kosi-Ulbl, Irena; Palacios, Ángel Rodríguez; Vukman, Joso

doi:10.1007/s00605-021-01596-8

Cited by 2 publications

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(iv) (Han [35]) F = K, A is a standard operator algebra on X , and D is a ring derivation. (v) (Vukman [68], see also [44]) F = K, A is a standard operator algebra on X , and D is linear. Clearly, in this case, the conclusion in the theorem remains true if the restriction that X is infinite-dimensional is removed.…”

Section: Introductionmentioning

confidence: 99%

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

Palacios

García

2023

European Journal of Mathematics

Self Cite

View full text Add to dashboard Cite

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 $$A\rightarrow AB-BA$$ A → A B - B A 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) $$H^*$$ H ∗ -algebra with no nonzero finite-dimensional direct summand are linear and continuous.

show abstract

Section: Introductionmentioning

confidence: 99%

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

Palacios

García

2023

European Journal of Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

Derivations characterized by monomials x2n in prime rings

Chang

Liu

2023

J. Algebra Appl.

View full text Add to dashboard Cite

Let [Formula: see text] be a prime ring of [Formula: see text] or [Formula: see text] and let [Formula: see text] be an additive map such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the maximal symmetric ring of quotients of [Formula: see text]. It is shown that there exist a derivation [Formula: see text] and an additive map [Formula: see text] with [Formula: see text] for all [Formula: see text], such that [Formula: see text]. This result is a natural generalization of the classic theorem of Herstein for Jordan derivations on prime rings. Moreover, it gives a purely algebraic version of the theorem recently obtained by Kosi-Ulbl, Rodriguez and Vukman for standard operator algebras on Banach spaces.

show abstract

A generalization of a theorem of Chernoff on standard operator algebras

Cited by 2 publications

References 16 publications

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

Derivations characterized by monomials x2n in prime rings

Contact Info

Product

Resources

About