At the regional conference held at the University of California, Irvine, in 1985 [24],
Harald Upmeier posed three basic questions regarding 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*-algebras and by Upmeier [23]
for JB-algebras. Question 2 was answered by Sakai [18] and Kadison [12] for von
Neumann algebras and by Upmeier [23] for JW-algebras. Question 3 was answered
by Upmeier [23] for JB-algebras, and it follows trivially from the Kadison–Sakai
answer to question 2 in the case of C*-algebras.
In the ternary case, both question 1 and question 3 were answered by Barton and
Friedman in [3] for complex JB*-triples. In this paper, we consider question 2 for
real and complex JBW*-triples and question 1 and question 3 for real JB*-triples.
A real or complex JB*-triple is said to have the inner derivation property if every
derivation on it is inner. By pure algebra, every finite-dimensional JB*-triple has the
inner derivation property. Our main results, Theorems 2, 3 and 4 and Corollaries 2
and 3 determine which of the infinite-dimensional real or complex Cartan factors
have the inner derivation property.