Jordan Ideals Revisited

Jordan homomorphisms revisited

“…We claim that if (i) holds, then we may assume without loss of generality that θ is bijective. Indeed, J = ker θ is a closed Jordan ideal of A and as such it is an ideal [9,Theorem 5.3] (see also [7] for an alternative proof). Thus A/J is a C * -algebra, and θ induces a Jordan isomorphismθ from A/J onto B.…”

Lie and Jordan Ideals in Reflexive Algebras

“…The converse problem, which has been studied for more than 50 years, is to describe Jordan ideals in terms of associative ideals, and to find conditions under which a non-zero Jordan ideal is in fact an associative ideal, or at least it contains a non-zero ideal. For a few references, see [2,3,27,12,13,28,29,7] and the literature therein. Among other papers, Oliveira in [27] showed that a weakly closed Jordan ideal in a nest algebra, which satisfies certain conditions, is an associative ideal.…”

“…Also, the authors in [2] showed that if the algebra generated by commutators of A is equal to A then each Jordan ideal is an associative ideal. In this section, we shall present some examples which shows that the Lie ideals and the Jordan ideals may be trivial even if the algebra does not satisfy those conditions.…”

On Jordan ideals and submodules: Algebraic and analytic aspects

“…The fundamental papers by Herstein [4] and Jacobson-Rickart [5] initiated the study of conditions guaranteeing that an algebra is JIF. It was proved that the class of JIF-algebras includes all simple algebras [4], all full matrix algebras M n (B), where B is a unital algebra [5], all W * -algebras (this was established in [3] for properly infinite W * -algebras and in [1], as well as in Bunce's unpublished paper, for the general case), and many other important examples. Denote by Alg([A , A ]) (respectively, by Id([A , A ])) the subalgebra (respectively, the ideal) of A generated by all commutators [a, b] = ab − ba, a, b ∈ A .…”

“…Denote by Alg([A , A ]) (respectively, by Id([A , A ])) the subalgebra (respectively, the ideal) of A generated by all commutators [a, b] = ab − ba, a, b ∈ A . It was shown in [1] that the class of JIF-algebras contains all algebras generated by commutators, i.e., satisfying the condition…”