Despite the rapid advances in Jordan theory and its diverse applications in the last two decades, there are few convenient references in book form for beginners and researchers in the field. This book is a modest attempt to fill part of this gap.

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.

Although the concept of a Jordan algebra was introduced originally for quantum formalism, by P. Jordan, J. von Neumann and E. Wigner [64], unexpected and fruitful connections with Lie algebras, geometry and analysis were soon discovered. In the last three decades, many more applications of Jordan algebraic structures have been found. We expose some of these applications in this book. Needless to say, the choice of topics is influenced by the author's predilections, and regrettable omissions are inevitable if the length of the book is to be kept manageable. Nevertheless, an effort has been made to cover sufficient basic results and Jordan techniques to provide a handy reference.