This book summarizes (and acquaints the reader with) some methods of differential geometry that are applied to functional analysis, mainly operator theory. Homogeneous spaces are the focus of interest, i.e. the orbits of group actions endowed with the structure of smooth manifolds. These spaces appear in many settings in the theory of operators and operator algebras. Only a few books have been written on this topic and the present one is very valuable, in part because it presents in terse and clear form the recent results that have previously only been available as journal articles. The author also raises new ideas, e.g. the investigation of operator ideals from the point of view of Lie theory.

Part 1 of the book is an introduction to Lie theory in infinite dimensions. In part 2, geometry of homogeneous spaces is studied. In part 3, the orbits are presented as manifolds, where differential geometric structure carries a lot of operator theoretic information. Many questions concerning further development of this field are set forth. The author also suggests tools that can be used to approach the problems studied. The book is very well arranged. It brings new and fresh ideas and is therefore a challenge and encouragement to those interested in the field.

Reviewer:

jdr