CMA member Cláudio Fernandes wrote the book “Introdução às Algebras de Operadores“.

Abstract:

The names of F. Riesz, J. von Neumann, I. Gelfand and M. Naimark are associated with the area of Operator Algebras. At the end of the 20th century, the area gained impact and underwent such advances that it was associated with the award of two Fields medals: Alain Connes in 1982 and Vaughan Jones in 1990.

Currently, it continues to be a very active research area. The main objective of this book is to present an introduction to the theory of Operator Algebras covering the main topics in the area. It consists of 6 chapters entitled: Spectral Theory in Banach Algebras, Representations of Banach Algebras, Fundamentals of C* Algebras, Representations and Classes of C* Algebras, Dynamical Systems and C*- Crossed Products and Introduction to von Neumann Algebras.

Operator Algebras is an important area of Functional Analysis with links to other areas of Mathematics, namely Operator Theory, Dynamical Systems, Non-Commutative Geometry, Probability Theory, Numerical Analysis, as well as Mathematical Physics. The book will thus be useful to all those interested in these areas, only requiring basic knowledge of Functional Analysis at the level of a degree course to be read.

This book can be used in a basic semester course in Operator Algebras. Some of the topics that are covered in the book, namely the theory of invertibility in noncommutative Banach algebras and the theory of C*- crossed products, can however form the basis of an advanced course.