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.
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.
C*-algebras and W*-algebras by Sh™ichir™ 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.
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 draft of a book is about half devoted to the basics on finite von Neumann algebras and half devoted to more advanced topics.
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.
Notes on operator algebras by Jesse Peterson (unpublished notes)
My own set of notes on operator algebras, last updated in 2015.
• 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.
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).
Groups with the Haagerup property; Gromov's a-T-menability by Pierre-Alain Cherix et al. (2001)
A collection of expository articles revolving around 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)
Emphasis is placed on approximation properties such as amenability or non-property T.
Ergodic Theory via Joinings by Eli Glasner (2003)
Goes in depth into factors and joinings of ergodic systems.
Ergodic theory and semisimple groups by Robert J. Zimmer (1984)
Foundational results from the theory of ergodic actions of semisimple Lie groups.
Discrete subgroups of semisimple Lie groups by G. A. Margulis (1991)
An in-depth treatise on ergodic actions of semisimple algebraic groups.
• Specialized topics in operator algebras:
C*-algebras and finite-dimensional approximations by Nathanial P. Brown and Narutaka Ozawa (2008)
Excellent book for advanced students; contains wealth of info on nuclearity, exactness, and quasidiagonality.
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)
Introduction to Hochschild cohomology in von Neumann algebras.
Finite von Neumann algebras and Masas by Allan M. Sinclair and Roger R. Smith (2008)
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.
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 from the perspective of "noncommutative Banach space" theory.
Tensor products of C*-algebras and operator spaces by Gilles Pisier (2020)
Focus on tensor products and the Connes-Kirchberg problem.
Operator algebras and their modules by David P. Blecher and Christian Le Merdy (2005)
Comprehensive book on the general theory of operator algebras and their (bi-)modules.
• Some other textbooks related to operator algebras
Introduction to tensor products of Banach spaces by Ray Ryan (2002)
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.
From left to right: Dan-Virgil Voiculescu, Sorin Popa, myself, Thomas Sinclair, and Roy M. Araiza.
Collaboration distance tool - Are you collaborating on your first paper? Do you want to know what your Erdős number will be? (Mine is currently 4: Jesse Peterson - Sorin Popa - Roger Smith - Charles Kam-tai Chui - Paul Erdős).).
Former PhD 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.
This photo includes some of my postdocs at the same workshop.
Top row: Bat-Od Battseren, Changying Ding, Sayan Das, myself, Kate Jushenko, Ben Hayes, Krishnendu Khan.
Bottom row: Kai Toyosawa, Dumindu DeSilva, Ishan Ishan, Srivatsav Kunnawalkam Elayavalli, and Thomas Sinclair.
