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Bounded derivations of \(JB^*\)-triples. (English) Zbl 0728.46046
A JB\({}^*\)-tripleis a complex Banach Space A together with a function \(\{.,.,.\}:A^ 3\to A\) (the triple product) which is continuous, symmetric in the outer variables and conjugate linear in the middle variable satisfying \[ (1)\quad D(a,b)D(x,y)-D(x,y)D(a,b)=D(D(a,b)x,y)- D(x,D(b,a)y) \] for all a,b,x,y\(\in A\), where D(a,b) is the operator defined by \(D(a,b)x=\{abx\};\)
(2) for each \(a\in A\) the operator D(a,a) is hermitian, has nonnegative spectrum and \(\| D(a,a)\| =\| a\|^ 2.\)
Each \(C^*\)-algebra (resp. \(JB^*\)-algebra) is a \(JB^*\)-triple with respect to the triple product \(\{xyz\}=1/2(xy^*z+zy^*x)\) (resp. \(\{xyz\}=(x\circ y^*)\circ z-(z\circ x)\circ y^*+(y^*\circ z)\circ x).\)
A derivation of A is a linear map \(\delta\) defined on a subtriple D(\(\delta\)) satisfying \(\delta \{xyz\}=\{\delta x,y,z\}+\{x,\delta y,z\}+\{x,y,\delta z\}\). Denote by Der(A) the bounded derivations of A and by Inder(A) the inner derivations i.e. finite sums of maps of the form \(\delta (a,b)x=\{abx\}-\{bax\}.\)
The main results of the present paper can be formulated as follows:
1) Each everywhere defined derivation of a JB-triple is bounded.
2) For every JB-Triple A Inder(A) is dense in Der(A) with respect to the strong operator topology.

MSC:
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
46L70 Nonassociative selfadjoint operator algebras
17C65 Jordan structures on Banach spaces and algebras
46H70 Nonassociative topological algebras
46L60 Applications of selfadjoint operator algebras to physics
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