Full Subalgebras of Jordan-Banach Algebras and Algebra Norms on JB ∗ -Algebras

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.

Nonassociative unitary Banach algebras

“…It follows from Lemma 3.1 that real non-commutative JB * -algebras whose Banach space is reflexive have a unit, and are Hilbertizable. It is known that the topology of any algebra norm on a JB * -algebra is stronger than that of the JB * -norm [24,Theorem 10]. Keeping in mind this result and Lemma 3.1, we can argue as in the proof of [1, Lemma 5.2] to obtain the following.…”

A Kaplansky theorem for JB*-triples

“…Every JB * -algebra has MOANT (compare [17,Theorem 10]). We shall say that a normed Jordan triple E has minimality of triple norm topology (MOTNT) if any other (not necessarily complete) triple norm dominated by the norm of E defines an equivalent topology.…”

“…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 not 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's theorem to the more general setting of JB * -algebras. The results established in [3] and [17] show that every not necessarily continuous Jordan monomorphism from a JB * -algebra to a normed Jordan algebra is bounded below. This result was proved again by S. Hejazian and A. Niknam in [11].…”

Banach Algebras With Large Groups of Unitary Elements

“…That A is unitary follows from Lemma 7.9 and [52]. Then, that A is strongly uniquely maximal follows from Corollary 7.5 and the fact that A has minimality of the norm [41,Proposition 11].…”