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

Abstract

Abstract.We introduce normed Jordan ß-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 g-algebras onto semisimple Jordan-Banach algebras with minimality of norm topology are continuous. As a consequence, the topology of the norm of a 75*-algebra is the smallest normable topology making the product continuous, and /¿»"-algebras have minimality of the norm. Some applications to (associative) C*-algebras are also given: (i) the associative normed algebras that are ranges of continuous (resp. contractive) Jordan homomorphisms from C* -algebras are bicontinuously (resp. isometrically) isomorphic to C*-algebras, and (ii) weakly compact Jordan homomorphisms from C*-algebras are of finite rank.

A Kaplansky theorem for JB*-triples

“…[8, Lemma 5.3]), equivalently, every C * -algebra has MOANT. It follows as a consequence of [3, Theorem 1] or [17,Theorem 10] or [11], that JB *algebras have MOANT. In the setting of (complex) JB * -triples, K. Bouhya and A. Fernández López proved the following result: Proposition 5.…”

“…Equivalently, if A denotes a real or complex C * -algebra (resp., a real or complex JB * -algebra) every continuous (triple) monomorphism T from A to a Banach algebra (resp., a Jordan Banach algebra) is bounded below. C * -algebras and JB * -algebras satisfy a stronger property: when A is a C * -algebra (resp., a JB * -algebra) every non-necessarily continuous monomorphism from A to a Banach algebra (resp., a Jordan Banach algebra) is bounded below (compare [8,Theorem 5.4] and [3,Theorem 1] or [17,Theorem 10] or [11]).…”

“…The arguments presented by A. Rodríguez Palacios in [22] were adapted by A. Bensebah [3] and J. Pérez, L. Rico and A. Rodríguez Palacios [17] to extend Kaplansky theorem to the more general setting of JB * -algebras. The results established in [3] and [17] show that every non necessarily continuous Jordan monomorphism from a JB * -algebra to a normed Jordan algebra is bounded below. This result was latter re-proved by S. Hejazian and A. Niknam in [11].…”