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Proceedings of the American Mathematical Society

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Full subalgebras of Jordan-Banach algebras and algebra norms on $\textrm {JB}^ *$-algebras

Authors: J. Pérez, L. Rico and A. Rodríguez
Journal: Proc. Amer. Math. Soc. 121 (1994), 1133-1143
MSC: Primary 46H70; Secondary 17C65, 46L70
MathSciNet review: 1195486
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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 $J{B^ \ast }$-algebra is the smallest normable topology making the product continuous, and $J{B^ \ast }$-algebras have minimality of the norm. Some applications to (associative) ${C^ \ast }$-algebras are also given: (i) the associative normed algebras that are ranges of continuous (resp. contractive) Jordan homomorphisms from ${C^ \ast }$-algebras are bicontinuously (resp. isometrically) isomorphic to ${C^ \ast }$-algebras, and (ii) weakly compact Jordan homomorphisms from ${C^ \ast }$-algebras are of finite rank.

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Article copyright: © Copyright 1994 American Mathematical Society