no. 3
On the CR-structure of certain linear group orbits in infinite dimensions
Kaup, Wilhelm1
1 Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen, Germany
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 535-554.

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits M explicitly and show as main result that every continuous CR-function on M has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite dimensions results from a recent joint paper with D. Zaitsev in Inventiones math. 153, 45-104.

Classification : 32V25, 17C50, 32H02, 32E20, 32M15, 46G20
Kaup, Wilhelm 1

1 Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen, Germany 
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Kaup, Wilhelm. On the CR-structure of certain linear group orbits in infinite dimensions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 535-554. http://www.numdam.org/item/ASNSP_2004_5_3_3_535_0/

[1] M. S. Baouendi - P. Ebenfelt - L. P. Rothschild, “Real Submanifolds in Complex Spaces and Their Mappings”, Princeton Math. Series 47, Princeton Univ. Press, 1998. | MR | Zbl

[2] M. S. Baouendi - F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), 387-421. | MR | Zbl

[3] A. Boggess, “CR Manifolds and the Tangential Cauchy-Riemann Complex”, Studies in Advanced Mathematics, CRC Press. Boca Raton, Ann Arbor, Boston, London 1991. | MR | Zbl

[4] N. Boubaki, “Integration”, Hermann, Paris 1965.

[5] S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Berlin-Heidelberg-New York, Springer, 1999. | MR | Zbl

[6] J. Faraut - L. Bouattour, Enveloppes polynômiales d'ensembles compacts invariants, Math. Nachr. 266 (2004), 20-26. | MR | Zbl

[7] T. Franzoni - E. Vesentini, “Holomorphic Maps and Invariant Distances”, North Holland, Amsterdam, 1980. | MR | Zbl

[8] L. A. Harris - W. Kaup, Linear algebraic groups in infinite dimensions, Illinois J. Math. 21 (1977), 666-674. | MR | Zbl

[9] W. Kaup, Algebraic Characterization of Symmetric Complex Banach Manifolds, Math. Ann. 228 (1977), 39-64. | MR | Zbl

[10] W. Kaup, A Riemann Mapping Theorem for Bounded Symmetric Domains in Complex Banach Spaces, Math. Z. 183 (1983), 503-529. | MR | Zbl

[11] W. Kaup, On spectral and singular values in JB * -triples, Proc. Roy. Irish. Acad. 96A (1996), 95-103. | MR | Zbl

[12] W. Kaup, Bounded symmetric domains and polynomial convexity, Manuscripta Math. 114 (2004), 391-398. | MR | Zbl

[13] W. Kaup - D. Zaitsev, On the CR-structure of compact group orbits associated with bounded symmetric domains, Invent. Math. 153 (2003), 45-104. | MR | Zbl

[14] O. Loos, “Jordan pairs”, Springer Lecture Notes 460, 1975. | MR | Zbl

[15] C. Sacré, Enveloppes polynomiales de compacts, Bull. Sci. Math. 116 (1992), 129-144. | MR | Zbl

[16] J. Sauter, “Randstrukturen beschränkter symmetrischer Gebiete”, Dissertation, Tübingen, 1995.