This page contains a collection of resources for students who are studying operator algebras. Some of these tools can be of use to any graduate students in mathematics, while other tools are more specific to operator algebra students, and much of this information is based on my personal tastes within operator algebras, e.g., von Neumann algebras, operator algebra connections to group theory, ergodic theory, etc.
Tools for research in operator algebras
ArXiv - A preprint server for mathematics, and other sciences. An indispensable tool for any mathematician. A basic way to keep up to date on research is to check the recent papers in the Operator Algebras and Group Theory sections on a regular basis.
MathSciNet - A great resource for looking up articles. A nice feature is that you can also find related articles since for each article it shows all of the recent articles that have cited it. The only problem is that it is behind a paywall. Although most universities have a subscription.
OASIS - A recently set up website containing useful resources for operator algebraists. The website contains a directory of operator algebraists, a listing of common journals that accept operator algebras articles, job opportunities, and seminars/conferences. This is a very useful website. I hope that it will continue to be maintained.
Some books I like in operator algebras and related fields:
• Background material on functional analysis and first courses in operator algebras:
Methods of modern mathematical physics I: Functional analysis; Revised and enlarged edition by Michael Reed and Barry Simon (1980)
An excellent textbook on functional analysis. I've used this book at least a couple of times when I've taught the course. A course in functional analysis by John B. Conway (1997)
A standard reference for the foundational material needed from functional analysis. I've used this book a couple of times when I've taught the course. This is a book I learned from when I was a graduate student. A course in operator theory by John B. Conway (2000)
Contains some of the basics of operator theory and operator algebras. On the operator algebras side of things it focuses on C*-algebras. This is also a book I learned from when I was a graduate student. Analysis now by Gert K. Pedersen (1988)
An introduction to real and functional analysis from an operator algebraic perspective. There is a second edition from 2012.
• Introductions to C* and von Neumann algebras:
C*-algebras by example by Kenneth R. Davidson (1996)
A very pleasant book starting at the basics of C*-algebra theory and culminating in Brown-Douglas-Fillmore theory. Emphasis is placed on analyzing a few examples in great depth. C*-algebras by Jacques Dixmier (1977)
An English translation of the 1969 French version. A classic textbook on C*-algebras, their representations, and the connections to representations of locally compact groups. This is still an excellent resource for these results. Von Neumann algebras by Jacques Dixmier (1981)
A classic textbook on von Neumann algebras. This contains much of the basic abstract theory of von Neumann algebras. Contains a complete discussion of reduction theory. Lectures on von Neumann algebras by Serban Stratila and Laszlo Zsido (1979)
A great introduction to the general theory of von Neumann algebras. This book contains an extensive bibliography containing nearly every article in operator algebras at the time of its publication. This is an English translation of the 1975 book in Romanian. A new English addition was published in 2019.
An introduction to operator algebras by Kehe Zhu (1993)
This gives a concise introduction to the basics of operator algebras and von Neumann algebras. I like the approach the author takes here and I've used this book before as a textbook for (half of) a course in operator algebras. I just wish that the book contained more results. It ends with comparison of projections, and even this could be expanded, e.g., there is no proof that the supremum of two finite projections is again finite. C*-algebras and their automorphism groups by Gert K. Pedersen (2018)
The second edition of an introductory book on C* and von Neumann algebras originally published in 1979. This book goes more in depth than most. It includes a discussion of crossed products coming from group actions.
An invitation to C*-algebras by William Arveson (1976)
A short bare-bones introduction to C*-algebras and their representations. Fundamentals of the theory of operator algebras Volumes I and II by Richard V. Kadison and John R. Ringrose (1983, 1986)
Contains all of the foundational material needed to study C* and von Neumann algebras. Operator algebras; Theory of C*-algebras and von Neumann algebras by Bruce Blackadar (2006)
This Encyclopedia of Mathematics volume contains a ton of information about the general theory of C* and von Neumann algebras. Most of the proofs are only sketched and so it is not the best textbook, but it makes for a great reference. Theory of Operator Algebras I, II, and III by M. Takesaki (2003)
A comprehensive treatment of "noncommutative integration theory". Many results are stated in a very general form, making this a good resource if you need a general result that you cannot find in other textbooks, e.g., if you need to use the non-commutative Lusin's theorem for a non-separable von Neumann algebra. The first volume is a revised version of Takesaki's 1980 textbook.
C*-algebras and W*-algebras by Shôichirô Sakai (1997)
An introduction to the basic theory of C* and von Neumann algebras.
Lectures on von Neumann algebras by David M. Topping (1971)
A brief text, introducing the basic properties of von Neumann algebras. Chapter 10 is devoted to the generator problems and presents the 1969 works of Douglas and Percy, and of Wogen.
C*-algebras and operator theory by Gerard J. Murphy (1990)
An introduction to the basic theory of C* and von Neumann algebras. Ends with a nice introduction to K-theory.
An introduction to II1 factors by Claire Anantharaman and Sorin Popa (preprint)
This (essentially complete) draft of a book is about half devoted to the basics on finite von Neumann algebras and half devoted to more advanced topics including amenability, property (T), and the Haagerup property. The book also contains an introduction to Popa's Intertwining-by-bimodules technique, and contains an example of a II1 factor with trivial fundamental group. This is an invaluable resource for anyone who wants to learn deformation/rigidity theory. Von Neumann algebras by Vaughan Jones (unpublished notes from 2009)
Incomplete notes on von Neumann algebras. This manuscript presents Haagerup's approach to Tomita-Takesaki theory in Chapter 13, which is perhaps the most elegant and concise approach. There is a nice discussion of the CAR and CCR algebras. There is also a proof on restriction of the index for subfactors.
Notes on operator algebras by Jesse Peterson (unpublished notes)
My own continually evolving set of notes on operator algebras, last updated in 2015. This starts out with the basics of Banach algebras and goes on from there. I also include some discussion on approximation properties of groups. Eventually I would like to add more advanced topics in the theory of II1 factors.
• Introductory books on group theory connected to operator algebras:
A course in abstract harmonic analysis by Gerald B. Folland (1995)
An excellent treatment of the foundational ideas in abstract harmonic analysis. This textbook is largely self-contained and is appropriate for a graduate student who has only had a course in real analysis. Although having a first course in functional analysis first would help. A second addition was published in 2015. Kazhdan's property (T) by Bachir Bekka, Pierre de la Harpe and Alain Valette (2008)
The standard resource for learning about Kashdan's property (T). This book also contains an extensive appendix coving basic topics in abstract harmonic analysis such as functions of positive type, induced representations, the Fell topology, and amenability. There is an essentially complete preprint available. Groups with the Haagerup property; Gromov's a-T-menability by Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette (2001)
More of a collection of expository articles than a textbook. Each chapter is written by a different subset of the author list, and revolves around a different aspect of the Haagerup property for locally compact groups.
• Some books on abstract ergodic theory:
Ergodic Theory; Independence and Dichotomies by David Kerr and Hanfeng Li (2016)
Another great introduction to the theory of ergodic actions of abstract discrete groups. Emphasis is placed on approximation properties such as amenability or non-property T. This is book is the main resource for learning sofic entropy, about half of the book is devoted to this topic. Ergodic Theory via Joinings by Eli Glasner (2003)
A great introduction to the theory of ergodic actions of abstract discrete groups. This book goes in depth into factors and joinings of ergodic systems, and proves the Furstenberg-Zimmer structure theorem. There is also a detailed discussion of entropy for ergodic actions of the integers. Ergodic theory and semisimple groups by Robert J. Zimmer (1984)
Many foundational results are explained from the theory of ergodic actions of semisimple Lie groups. Topics include the Howe-Moore theorem, amenability, property (T), Margulis's normal subgroup theorem, Margulis's superrigidity theorem, and Zimmer's cocycle extension of Margulis's superrigidity theorem.
Discrete subgroups of semisimple Lie groups by G. A. Margulis (1991)
An in-depth treatise on ergodic actions of semisimple algebraic groups, and their lattices. There is some overlap of topics between this book and Zimmer's ergodic theory book. This book usually contains the most general form of the result. As a consequence, this book also has significantly more preliminary results that are needed in order to obtain this level of generality.
• Specialized topics in operator algebras:
C*-algebras and finite-dimensional approximations by Nathanial P. Brown and Narutaka Ozawa (2008)
An excellent book for the advanced student who has already had a course in operator algebras. This contains a wealth of information on advanced topics in the theory of C*-algebras. Nuclearity, exacness, and quasidiagonality are all discussed in detail. There are also a number of approximation properties for groups that are discussed in detail, including amenability, the haagerup property, weak amenability, exactness, the complete metric approximation property, (non) property (T), biexactness. Applications are given to the classification of group von Neumann algebras. Hilbert C*-modules, a toolkit for operator algebraists by E.C. Lance (1995)
The standard reference for Hilbert C*-modules. Hochschild cohomology of von Neumann algebras by Allan M. Sinclair and Roger R. Smith (1995)
An introduction to Hochschild cohomology in von Neumann algebras. Contains vanishing results for completely bounded Hoschschild cohomology and uses this to show vanishing results for von Neumann algebras with property (Gamma), or containing a Cartan subalgebra. Finite von Neumann algebras and Masas by Allan M. Sinclair and Roger R. Smith (2008)
This book has a brief introduction to finite von Neumann algebras, including comparison of projections, and subfactors and the basic construction, but the majority of the book is devoted to studying masas (maximal abelian von Neumann subalgebras).
• Books on operator spaces
Operator spaces by Edward G. Effros and Zhong-Jin Ruan (2000)
An introduction to operator spaces from two of the pioneers in the field. This book takes a "noncommutative Banach space" approach and covers the basic theory and some advanced topics.
Completely bounded maps and operator algebras by Vern Paulsen (2002)
An elegantly written treatise on operator spaces and operator algebras. Introduction to operator space theory by Gilles Pisier (2003)
An introduction to operator spaces from the perspective of "noncommutative Banach space" theory. Tensor products of C*-algebras and operator spaces; The Connes-Kirchberg problem by Gilles Pisier (2020)
An introduction to operator spaces with an emphasis on tensor products and the Connes-Kirchberg problem (for which a solution has recently been announced). Operator algebras and their modules by David P. Blecher and Christian Le Merdy (2005)
This is a comprehensive book on the general theory of operator algebras and their (bi-)modules. This is a great reference book to have if you need to work with operator space (bi-)modules.
• Some other textbooks related to operator algebras
Introduction to tensor products of Banach spaces by Ray Ryan (2002)
Unlike with Hilbert spaces, there are multiple different norms that one can put on the tensor product of Banach spaces so as to get a normed space. A similar phenomenon happens with C*-algebras. This book makes for a good reference to keep track of the different tensor products and the properties they satisfy. Classical descriptive set theory by Alexander S. Kechris (1995)
A fantastic source for learning about the basics of descriptive set theory.
Five generations attending the workshop "Approximation properties in operator algebras and ergodic theory" at the Institute of Pure and Applied Mathematics in May of 2018.
Former students (all at Vanderbilt University) Some of my PhD students attending the "Eighteenth Annual Spring Institute on Noncommutative Geometry and Operator Algebras" at Vanderbilt University in May of 2023.
From left to right: Kai Toyosawa, Dumindu DeSilva, Changying Ding, Sayan Das, myself, Krishnendu Khan, Thomas Sinclair, Ishan Ishan, and Srivatsav Kunnawalkam Elayavalli.
Current and former postdocs mentored This photo includes some of my postdocs at the same workshop.
Top row, left to right: Bat-Od Battseren, Changying Ding, Sayan Das, myself, Kate Jushenko, Ben Hayes, Krishnendu Khan.
Bottom row, left to right: Kai Toyosawa, Dumindu DeSilva, Ishan Ishan, Srivatsav Kunnawalkam Elayavalli, and Thomas Sinclair.
