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.
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