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Bounded symmetric domains in Banach spaces. (English) Zbl 1473.46004

Hackensack, NJ: World Scientific (ISBN 978-981-12-1410-3/hbk; 978-981-12-1412-7/ebook). xi, 393 p. (2021).
The book under review focuses on infinite dimensional bounded symmetric domains and some recent advances in the geometric and analytic theory of these domains.
While Lie theory has been an important tool in the investigation of finite dimensional bounded symmetric domains, Jordan algebras and Jordan triple systems have gradually become a significant part of the theory of bounded symmetric domains since they provide a unified treatment in both the finite and infinite dimensional settings. In this book, the author presents an introduction to the basic theory of infinite dimensional Jordan and Lie algebras and explains how they are used to show that a bounded symmetric domain is biholomorphic to the open unit ball of a Banach space with a Jordan structure.
Chapter 1 contains some basic concepts and notation in complex analysis in Banach spaces and a brief review of Banach manifolds. Symmetric Banach manifolds, which generalize finite dimensional Hermitian symmetric spaces, are introduced, together with the associated Lie structures.
Chapter 2 discusses Jordan and Lie structures, including the structures of Jordan algebras and Jordan triple systems, the Tits-Kantor-Koecher construction, \(\textup{JB}^\ast\)-triples and Cartan factors.
Chapter 3 gives a full discussion of bounded symmetric domains. After discussing the Jordan and Lie structures of symmetric Banach manifolds, the author proves one of the most important theorems in the book due to W. Kaup [Math. Appl., Dordr. 303, 204–214 (1994; Zbl 0810.46075)], i.e., that a domain in a complex Banach space is a bounded symmetric domain if and only if it is biholomorphic to the open unit ball of a \(\textup{JB}^\ast\)-triple. This theorem provides a Jordan approach to the geometry of bounded symmetric domains, which relies on two fundamental bounded linear operators on a \(\textup{JB}^\ast\)-triple, namely, the Bergman operator and a left multiplication called box operator. In Sections 3.3–3.5, the author gives a classification of finite-rank bounded symmetric domains in terms of \(\textup{JB}^\ast\)-triples, studies the boundary structures of a bounded symmetric domain and its boundary components in terms of Jordan structures, discusses invariant metrics, the Schwarz lemma and the iteration of holomorphic self-maps on a bounded symmetric domain, in particular, the Denjoy-Wolff theorem. In Sections 3.6–3.8, the author studies when bounded symmetric domains are holomorphically equivalent to Siegel domains, which bounded symmetric domains are holomorphic homogeneous regular domains, and discusses a Jordan approach to the classification of bounded symmetric domains.
Chapter 4 touches on various topics in function theory on bounded symmetric domains, including distortion theorems, Bloch functions and the Bloch constant, and composition operators on Banach spaces of Bloch functions.
This book has a concise bibliography, and it can serve as a convenient reference for a broad readership including research students.

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
46G20 Infinite-dimensional holomorphy
17C65 Jordan structures on Banach spaces and algebras
46L70 Nonassociative selfadjoint operator algebras

Citations:

Zbl 0810.46075
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