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A Kaplansky Theorem for JB*-triples

Francisco J. Fernández-Polo, Jorge J. Garcés, Antonio M. Peralta

Abstract

Let $T:E\rightarrow F$ be a non-necessarily continuous triple homomorphism from a (complex) JB$^*$-triple (respectively, a (real) J$^*$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$^*$-algebras and of A. Bensebah and J.Pérez, L. Rico and A. Rodr'\iguez Palacios in the setting of JB$^*$-algebras.

A Kaplansky Theorem for JB*-triples

Abstract

Let be a non-necessarily continuous triple homomorphism from a (complex) JB-triple (respectively, a (real) JB-triple) to a normed Jordan triple. The following statements hold: (1) has closed range whenever is continuous (2) has closed range whenever is continuous This result generalises classical theorems of I. Kaplansky and S.B. Cleveland in the setting of C-algebras and of A. Bensebah and J.Pérez, L. Rico and A. Rodr'\iguez Palacios in the setting of JB-algebras.
Paper Structure (3 sections, 16 theorems, 43 equations)

This paper contains 3 sections, 16 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Minimality of norm topology for JB$^*$-triples
  3. Separating spaces for triple homomorphisms

Key Result

Theorem 2

Let $A$ be a commutative C$^*$-algebra with a norm $\|.\|$ and let $\|.\|_1$ be another norm on $A$ under which $A$ is a normed algebra. Then $\|a\| \leq \|a\|_1$, for every $a$ in $A$. Further, for any algebra norm, $\| .\|_{1}$, on $A_{sa}$, the inequality $\|a\| \leq \|a\|_1$ holds for every $a$

Theorems & Definitions (27)

  • Remark 1
  • Theorem 2
  • Lemma 3
  • proof
  • Remark 4
  • Proposition 5
  • Proposition 6
  • proof
  • Corollary 7
  • Corollary 8
  • ...and 17 more