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

Abstract

We introduce normed Jordan Q -algebras, namely, normed Jordan algebras in which the set of quasi-invertible elements is open, and we prove that a normed Jordan algebra is a Q -algebra if and only if it is a full subalgebra of its completion. Homomorphisms from normed Jordan Q -algebras onto semisimple Jordan-Banach algebras with minimality of norm topology are continuous. As a consequence, the topology of the norm of a J B ∗ J{B^ \ast } -algebra is the smallest normable topology making the product continuous, and J B ∗ J{B^ \ast } -algebras have minimality of the norm. Some applications to (associative) C ∗ {C^ \ast } -algebras are also given: (i) the associative normed algebras that are ranges of continuous (resp. contractive) Jordan homomorphisms from C ∗ {C^ \ast } -algebras are bicontinuously (resp. isometrically) isomorphic to C ∗ {C^ \ast } -algebras, and (ii) weakly compact Jordan homomorphisms from C ∗ {C^ \ast } -algebras are of finite rank.