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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
##### Keywords:
inner derivation; $$JB^ *$$-triple; triple product
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