Complex Jordan Algebras

Abstract

Abstract In later chapters we will study complex tube domains over symmetric cones. These are domains in the complexifications of Euclidean Jordan algebras. The first two sections of the present chapter discuss the relevant definitions and basic facts, including some properties of the structure group of a complexified Euclidean Jordan algebra which will be needed later. The subsequent sections will not be needed later in this book; however, they contain basic background information. Section 3 gives a generalization of the Jordan canonical form of matrices, which is valid in any complex Jordan algebra. In Section 4 we prove that the class of Euclidean Jordan algebras coincides with the class of formally real Jordan algebras. In Section 5 we prove that the complexifications of Euclidean Jordan algebras exhaust the seemingly larger class of complex semi-simple Jordan algebras.