##
**Jordan structures in geometry and analysis.**
*(English)*
Zbl 1238.17001

Cambridge Tracts in Mathematics 190. Cambridge: Cambridge University Press (ISBN 978-1-107-01617-0/hbk). x, 261 p. (2012).

Jordan systems: algebras, triples and pairs, play an important role in some aspects of geometry in finite dimension (see, for instance, the monograph by O. Loos [Bounded symmetric domains and Jordan pairs. Mathematical Lectures, University of California, Irvine (1977)], and the references therein). In the monograph under review, the author exposes recent advances of Jordan systems in not necessarily finite-dimensional geometry, as well as in functional analysis.

Quoting from the introduction: “The aim of the book is to introduce to a wide readership, including research students, the close connections between Jordan algebras, geometry, and analysis. In particular, we give a self-contained and systematic exposition of a Jordan algebraic approach to symmetric manifolds which may be infinite-dimensional, and some fundamental results of Jordan theory in complex and functional analysis. In short, this book is about Jordan geometric analysis.”

The monograph is split into three chapters. The first one deals with the basic definitions of Jordan algebras and triple systems, and with the important Tits-Kantor-Koecher construction of a \(3\)-graded Lie algebra attached to any such Jordan system. The material of the second chapter is centered around Riemannian symmetric spaces and Jordan theory, in the setting of Banach manifolds and Banach Lie groups, and its relationship with Jordan algebras and triple systems. It covers material developed by many authors, mainly by Kaup, Upmeier and the author. Finally, Chapter 3 deals with the basic properties of \(JB^*\)-triples, although it includes some new results on \(JH\)-triples, too.

Quoting from the introduction: “The aim of the book is to introduce to a wide readership, including research students, the close connections between Jordan algebras, geometry, and analysis. In particular, we give a self-contained and systematic exposition of a Jordan algebraic approach to symmetric manifolds which may be infinite-dimensional, and some fundamental results of Jordan theory in complex and functional analysis. In short, this book is about Jordan geometric analysis.”

The monograph is split into three chapters. The first one deals with the basic definitions of Jordan algebras and triple systems, and with the important Tits-Kantor-Koecher construction of a \(3\)-graded Lie algebra attached to any such Jordan system. The material of the second chapter is centered around Riemannian symmetric spaces and Jordan theory, in the setting of Banach manifolds and Banach Lie groups, and its relationship with Jordan algebras and triple systems. It covers material developed by many authors, mainly by Kaup, Upmeier and the author. Finally, Chapter 3 deals with the basic properties of \(JB^*\)-triples, although it includes some new results on \(JH\)-triples, too.

Reviewer: Alberto Elduque (Zaragoza)

### MSC:

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17C65 | Jordan structures on Banach spaces and algebras |

17C37 | Associated geometries of Jordan algebras |

46H70 | Nonassociative topological algebras |

53C35 | Differential geometry of symmetric spaces |

46K70 | Nonassociative topological algebras with an involution |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |