Lie and Jordan structures in Banach algebras

Mappings preserving zero products

“…Let I be the kernel of ϕ, which is a closed Jordan ideal of the C * -algebra A. By [14], I is also a twosided ideal of A. Then ϕ induces a Jordan isomorphism from the quotient algebra A/I onto B, which is again zero-product preserving by Lemma 4.14.…”

Lie and Jordan Ideals in Reflexive Algebras

“…The relationship between Lie ideals and similarity invariant subspaces has been investigated for more than 30 years. For a few references, see [17,24,25,16,26,7,13,12,3] and the literature therein. For the history and the development in this area, we refer to a good introduction given by Hopenswasser and Paulsen [16].…”

“…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.…”

Full subalgebras of Jordan-Banach algebras and algebra norms on 𝐽𝐵*-algebras

“…Let A be a C*-algebra and tp a Jordan homomorphism from A onto B under the assumptions in the statement. Since closed Jordan ideals of a C*-algebra are associative ideals (see [7,Theorem 5.3. ] or [21,Theorem 4.3]), A/ Ker(tp) is a C*-algebra and we may assume that tp is a one-to-one mapping.…”