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Weak continuity of holomorphic automorphisms in $$JB^*$$-triples. (English) Zbl 0812.46066
A complex Banach space $$E$$ becomes a $$JB^*$$-triple iff it is endowed with a Jordan triple product $$E\times E\times E\times E$$, $$(x,y,z)\mapsto\{xyz\}$$ which satisfies the conditions:
(J$$_1$$) $$\{xyz\}$$ is symmetric bilinear in the outer variables $$x$$, $$z$$ and conjugate linear in the inner variable $$y$$,
(J$$_2$$) $$\{ab\{xyz\}\}= \{\{abx\} yz\}+ \{xy\{abz\}\}- \{x\{bay\} z\}$$ (Jordan triple identity),
(J$$_3$$) The operator $$x\square x$$ defined by $$z\mapsto \{xxz\}$$ is a Hermitian operator with positive spectrum,
(J$$_4$$) $$\| xxx\|= \| x\|^ 3$$; this condition can be replaced by $$\| x\square x\|= \| x\|^ 2$$ and it is well- known that $$\| x\square y\|\leqq\| x\|$$. $$\| y\|$$ for all elements in $$E$$.
Every $$C^*$$-algebra $$A$$ becomes a $$JB^*$$-triple $$A^{JT}$$ in the triple product $$\{xyz\}= (xy^* z+ zy^* x)/2$$.
A $$JB^*$$-algebra is a Banach space with a Jordan product $$x\circ y$$ and a conjugate linear involution * such that
(J$$_4$$) $$\| x\circ x\|= \| x\|^ 2$$,
(J$$_5$$) $$\| x\circ y\|\leq\| x\|$$. $$\| y\|$$,
(J$$_6$$) $$(x\circ y)^*= y^*\circ x^*$$.
A $$C^*$$-algebra $$A$$ becomes a $$\text{JB}^*$$-algebra $$A^ J$$ in the Jordan product $$x\circ y= (xy+ yx)/2$$ and a $$JB^*$$-algebra $$B$$ becomes a $$JB^*$$-triple $$B^ T$$ in the triple product $\{xyz\}= x\circ (y^*\circ z)- y^*\circ(z\circ x)+ z\circ(x\circ y^*).$ It is easy to see that for any $$C^*$$-algebra $$A$$, $$(A^ J)^ T= A^{JT}$$.
In the paper under review, the authors study some problems of the geometry of Banach spaces underlying in these algebraic structures. It is interesting that: two complex Banach spaces are isometrically isomorphic if and only if the corresponding open unit balls are biholomorphically equivalent [the second author and H. Upmeier, Proc. Am. Math. Soc. 58, 129-133 (1976; Zbl 0337.32012)]. One reduces therefore some problems of studying Banach space geometry to studying automorphisms of unit balls.
The main problem the authors are interested in is the transitivity of the group $$\operatorname{Aut}(D)$$ of all biholomorphic automorphisms of the open unit ball in $$JB^*$$-triple. The main result they proved is: the set $$C=\text{cont}_ w(E)$$ of all $$a\in E$$, such that the $$a$$-squaring map $$q_ a: x\mapsto\{xax\}$$ on $$E$$ is weakly continuous on bounded sets, is a closed characteristic (triple) ideal in $$E$$ and $$g\in \text{Aut}(D)$$ is weakly continuous if and only if $$g(0)\in C$$. The elements of $$\text{cont}_ w(E)$$ are closely related to compact operators on Hilbert space.

##### MSC:
 46L70 Nonassociative selfadjoint operator algebras 17C65 Jordan structures on Banach spaces and algebras 46B20 Geometry and structure of normed linear spaces 46G20 Infinite-dimensional holomorphy
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