# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Barton, T.J., Dang, T., Horn, G.: Normal representations of Banach Jordan triple systems. Proc. Am. Math. Soc.102, 551–555 (1988) · Zbl 0661.46045 · doi:10.1090/S0002-9939-1988-0928978-2 [2] Barton, T.J., Friedman, Y.: Bounded derivations of JB*-triples. Q.J. Math., Oxf.41, 255–268 (1990) · Zbl 0728.46046 · doi:10.1093/qmath/41.3.255 [3] Barton, T.J., Timoney, R.W.: Weak* continuity of Jordan triple products and applications. Math. Scand.59, 177–191 (1986) · Zbl 0621.46044 [4] Calkin, J.W.: Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. Math.42, 839–873 (1941) · Zbl 0063.00692 · doi:10.2307/1968771 [5] Chu, C., Jochum, B.: The Dunford-Pettis property in C*-algebras. Studia Math.97, 59–64 (1990) · Zbl 0734.46034 [6] Cunningham, F., Effros, E.G., Roy, N.M.: M-Structure in dual Banach spaces. Isr. J. Math.14, 304–308 (1973) · Zbl 0285.46059 · doi:10.1007/BF02764892 [7] Dineen, S., Timoney, R.M.: The centroid of a JB*-triple system. Math. Scand.62, 327–342 (1988) · Zbl 0655.46044 [8] Friedman, Y., Russo, B.: The Gelfand-Naimark theorem for JB*-triples. Duke Math. J.53, 139–148 (1986) · Zbl 0637.46049 · doi:10.1215/S0012-7094-86-05308-1 [9] Harris, L.A., Kaup, W.: Linear algebraic groups in infinite dimensions. Ill. J. Math.21, 666–674 (1977) · Zbl 0385.22011 [10] Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Berlin Heidelberg New York: Springer 1965 · Zbl 0137.03202 [11] Horn, G.: Characterization of the predual and ideal structure of a JBW*-triple. Math. Scand.61, 117–133 (1987) · Zbl 0659.46062 [12] Isidro, J.M.: A glimpse at the theory of Jordan Banach triple systems. Rev. Mat. Univ. Complutense Madr.2, 145–156 (1989) · Zbl 0722.46024 [13] Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z.183, 503–529 (1983) · Zbl 0519.32024 · doi:10.1007/BF01173928 [14] Kaup, W.: Über die Klassifikation der symmetrischen Hermiteschen Mannigfaltigkeiten unendlicher Dimension I, II. Math. Ann.257, 463–483 (1981);262, 503–529 (1983) · Zbl 0482.32010 · doi:10.1007/BF01465868 [15] Kaup, W., Upmeier, H.: Banach spaces with biholomorphically equivalent unit balls are isomorphic. Proc. Am. Math. Soc.58, 129–133 (1976) · Zbl 0337.32012 · doi:10.1090/S0002-9939-1976-0422704-3 [16] Kaup, W., Upmeier, H.: Jordan Algebras and Symmetric Siegel Domains in Banach Spaces. Math. Z.157, 179–200 (1977) · Zbl 0357.32018 · doi:10.1007/BF01215150 [17] Loos, O.: Jordan pairs. (Lect. Notes Math., vol. 460) Berlin Heidelberg New York: Springer 1975 · Zbl 0301.17003 [18] Payá, R., Pérez, J., Rodriguez, A.: Noncommatative Jordan C*-algebras. Manuscr. Math37, 87–120 (1982) · Zbl 0483.46049 · doi:10.1007/BF01239948 [19] Rodriguez Palacios, A.: On the strong* topology of JBW*-triple. Q. J. Math., Oxf. (to appear) · Zbl 0723.17026 [20] Stachó, L., Isidro, J.M.: Algebraically compact elements in JBW*-triple systems. Acta Sci. Math.54, 171–190 (1990) · Zbl 0736.46053 [21] Takesaki, M.: Theory of Operator Algebras I. Berlin Heidelberg New York: Springer 1979 · Zbl 0436.46043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.