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Symmetric Banach manifolds and Jordan \(C^*\)-algebras. (English) Zbl 0561.46032
North-Holland Mathematics Studies, 104. Notas de Matemática, 96. Amsterdam - New York - Oxford: North-Holland. XII, 444 p. $ 57.75; Dfl. 150.00 (1985).
The book is a self-contained introduction to the theory of symmetric (complex) Banach manifolds which are natural generalizations of the classical (hermitian) symmetric spaces. If symmetric manifolds have additional structures (the existence of global symmetries and the homogeneity under holomorphic transformations) then they are completely characterized in terms of certain non associative generalizations of operator algebras - the so called Jordan \(C^*\)-algebras and Jordan triple systems. Using this Jordan algebraic description, many holomorphic and geometric properties of symmetric Banach manifolds can be interpreted algebraically and, conversely, the holomorphic structure associated with Jordan operator algebras can be useful for a deeper understanding of these algebras and their automorphism groups.
The book consists of two parts. Part I (Sections 1-13) is devoted to the theory of transformation groups on analytic Banach manifolds. The main purpose of this part is to endow certain groups of bianalytic transformations on Banach manifolds with the analytic structure of a Banach Lie group. In Part II (Sections 14-23) these results are applied to a systematic study of the special symmetric Banach manifolds and their algebraic characterization mentioned above in terms of Jordan algebras and Jordan triple systems.
There are a lot of examples illustrating the general theory, in particular the ”classical” symmetric Banach manifolds are considered in the last section 23.
The book is of great interest both for specialists in the theory of operator algebras since it gives a powerful instrument in the study of holomorphic functions and interesting and fruitful applications of Jordan operator algebras.
Reviewer: Sh.A.Ayupov

MSC:
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17C65 Jordan structures on Banach spaces and algebras
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
32M05 Complex Lie groups, group actions on complex spaces
32A99 Holomorphic functions of several complex variables
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces