Operator Algebras and Their Modules: An Operator Space Approach
This monograph is devoted to a study of operator algebras and their modules by means of operator space theory. The book is divided into eight chapters and an appendix. The first chapter contains basic facts on operator spaces and completely bounded maps. In the second chapter, a study of operator algebras begins. An operator algebra is just a closed subalgebra of the space B(H) of bounded linear operators on a Hilbert space H. Hence it has a Banach algebra structure and an operator space structure. This leads to the definition of an abstract operator algebra - it is a Banach algebra A with an operator space structure such that there exists a completely isometric homomorphism of A into some B(H). The rest of the chapter describes basic constructions of operator algebras. The third chapter contains an introduction to operator modules. An operator module is an operator space that is also A -module for a Banach algebra A and satisfies an additional property. Many examples and some subclasses (Hilbert modules, operator modules over operator algebras) are described. The remaining five chapters deal with more advanced or special topics. They include ‘extremal theory’ (noncommutative Shilov boundaries and related things), a completely isomorphic theory of operator algebras (in addition to the completely isometric one considered in the second chapter), tensor products of operator algebras, selfadjointness criteria and C*-modules. The book ends with an appendix containing basic facts from operator theory, Banach space theory, Banach algebras and C*-algebras.