Fundamentals of the Theory of Operator Algebras

Fundamentals of the Theory of Operator Algebras is a four-volume textbook on the classical theory of operator algebras written by Richard Kadison and John Ringrose. The first two volumes, published in 1983 and 1986, are entitled (I) Elementary Theory and (II) Advanced Theory; the latter two volumes, published in 1991 and 1992, present complete solutions to the exercises in volumes I and II.

Contents

• Volume I: Elementary Theory

Chapter 1. Linear spaces

Chapter 2. Basics of Hilbert Space and Linear Operators

Chapter 3. Banach Algebras

Chapter 4. Elementary C*-Algebra Theory

Chapter 5. Elementary von Neumann Algebra Theory

• Volume II: Advanced Theory

Chapter 6. Comparison Theory of Projection

Chapter 7. Normal States and Unitary Equivalence of von Neumann Algebras

Chapter 8. The Trace

Chapter 9. Algebra and Commutant

Chapter 10. Special Representation of C*-Algebras

Chapter 11. Tensor Products

Chapter 12. Approximation by Matrix Algebras

Chapter 13. Crossed Products

Chapter 14. Direct Integrals and Decompositions

Volumes III and IV follow Volumes I and II chapter-by-chapter with solutions to the exercises.

Reception

According to Nick Lord (writing for The Mathematical Gazette), the two volumes "met with immediate acclaim from functional analysts as a clear, careful, self-contained introduction to C*- and von Neumann algebra theory", something many in the field felt was missing.^[1] Ringrose and Kadison wrote with a pedagogical goal in mind, purposefully keeping the references sparse and including a long list of exercises.^{[2]: viii–x} These exercises have been described as "outstanding" and the solutions, in the later volumes, have been similarly commended.^[3] They wrote of their textbook that:

[Its] primary goal is to teach the subject and lead the reader to the point where the vast recent literature, both in the subject proper and in its many applications, becomes accessible. Although we have put major emphasis on making the material presented clear and understandable, the subject is not easy; no account, however lucid, can make it so.^[2]: vii

Gert K. Pederson wrote in 1994 of the volumes' popularity:

The theory [of operator algebras] has certainly come of age and has become fashionable. At the same time, the demand for concise textbooks that cover the subject has become noticeable [...] Fundamentals in the theory of operator algebras [...] have quickly established themselves as The Textbooks in Operator Algebra Theory. [...] The 715 exercises serve to illustrate and extend the results and examples in the text, but also help the reader to develop working techniques and facility with the subject matter. Some are routine, requiring nothing more than a clear understanding of a definition or a result for their solution. Other exercises (and groups of exercises) constitute small (guided) research projects. Anyone who claims to have solved them all is either boasting or is truly a master of our subject.^[4]

References

1. Lord, Nick (March 1998). "Review: Fundamentals of the Theory of Operator Algebras, Volume IV by R. V. Kadison, J. R. Ringrose". The Mathematical Gazette. 82 (493): 156–157. doi:10.2307/3620194. JSTOR 3620194.
2. Ringrose, John R.; Kadison, Richard V. (1983). Fundamentals of the Theory of Operator Algebras. Volume I: Elementary Theory. Academic Press.
3. O'Connor, John J.; Robertson, Edmund F. (June 1998), "John Robert Ringrose", MacTutor History of Mathematics Archive, University of St Andrews
4. O'Connor, John J.; Robertson, Edmund F. (March 2021), "John Ringrose's books", MacTutor History of Mathematics Archive, University of St Andrews