Derivations on Real and Complex JB*-Triples

At the regional conference held at the University of California,Irvine, in 1985 24], Harald Upmeier posed three basic questionsregarding derivations on JB*-triples: (1) Are derivations automatically bounded? (2) When are all bounded derivations inner? (3) Can bounded derivations be approximated by inner derivations? These three questions had all been answered in the binary cases.Question 1 was answered affirmatively by Sakai 17] for C*-algebrasand by Upmeier 23] for JB-algebras. Question 2 was answeredby Sakai 18] and Kadison 12] for von Neumann algebras andby Upmeier 23] for JW-algebras. Question 3 was answered byUpmeier 23] for JB-algebras, and it follows trivially fromthe Kadison–Sakai answer to question 2 in the case ofC*-algebras. In the ternary case, both question 1 and question 3 were answeredby Barton and Friedman in 3] for complex JB*-triples. In thispaper, we consider question 2 for real and complex JBW*-triplesand question 1 and question 3 for real JB*-triples. A real orcomplex JB*-triple is said to have the inner derivation propertyif every derivation on it is inner. By pure algebra, every finite-dimensionalJB*-triple has the inner derivation property. Our main results,Theorems 2, 3 and 4 and Corollaries 2 and 3 determine whichof the infinite-dimensional real or complex Cartan factors havethe inner derivation property.