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DERIVATIONS ON REAL AND COMPLEX JB*-TRIPLES

Published online by Cambridge University Press:  06 March 2002

TONY HO
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697-3875, USA; [email protected]
JUAN MARTINEZ-MORENO
Affiliation:
Departamento Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain; [email protected], [email protected]
ANTONIO M. PERALTA
Affiliation:
Departamento Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain; [email protected], [email protected]
BERNARD RUSSO
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697-3875, USA; [email protected]
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Abstract

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.

Type
Research Article
Copyright
2002 London Mathematical Society

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