A Kaplansky Theorem for JB*-triples

Let $T:E\to F$ be a non-necessarily continuous triple homomorphism from a (complex) JB${}^{\ast }$-triple (respectively, a (real) J${}^{\ast }$B-triple) to a normed Jordan triple. The following statements hold: (1) $T$ has closed range whenever $T$ is continuous (2) $T$ has closed range whenever $T$ is continuous This result generalises classical theorems of I. Kaplansky and S.B. Cleveland in the setting of C${}^{\ast }$-algebras and of A. Bensebah and J.Pérez, L. Rico and A. Rodr’\iguez Palacios in the setting of JB${}^{\ast }$-algebras.