manuscripta mathematica

, Volume 86, Issue 1, pp 311-335

First online:

On real forms of JB*-triples

  • José M. IsidroAffiliated withFacultad de Matemáticas, Universidad de Santiago
  • , W. KaupAffiliated withMathematisches Institut, Universität Tübingen
  • , Angel Rodríguez PalaciosAffiliated withDep. Analisis Matematico Facultad de Ciencias, Universidad de Granada

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We introduce real JB*-triples as real forms of (complex) JB*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB*-triples is an isometry if and only if it commutes with the cube mappinga→a 3={aaa}. This generalizes a result of Dang for complex JB*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB*-triple numerical invariants that are respected by surjective linear isometries.