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.
(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.
Reviewer: No Ngoc Diep (Hanoi)
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 |
Keywords:
Jordan triple product; Hermitian operator with positive spectrum; \(\text{JB}^*\)-algebra; \(C^*\)-algebra; Banach space geometry; compact operators on Hilbert spaceCitations:
Zbl 0337.32012References:
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