## MATH 582, Spring 2023

**Course Title:**Introduction to C*-algebra theory

**Instructor:**Nigel Higson, 228 McAllister Building

**Class Meetings:**Tuesdays and Thursdays, 4:35-5:50, in 106 McAllister Building.

**Office Hours:**By appointment (contact me in class or by email). We can meet either in my office or by Zoom.

**Overview:**C*-algebra theory is, or at least it began as, the study of relations among operators on Hilbert space. Of special importance are the varieties of ways in which specific operator relations may be realized on Hilbert space, and because of this there are obvious relations with group theory, via unitary group representations, and quantum theory, via the canonical commutation and anticommutation relations, among other things. I will try to develop the basic theory of C*-algebras in the class, roughly as follows:

- Hilbert spaces and projection operators. Adjoint and norm. Unitary operators, isometries and partial isometries. Spectrum of bounded and unbounded operators. Spectral theorem for compact operators.
- Commutative Banach algebras. Spectrum and spectral radius formula. Gelfand transform. Definition and characterization of commutative C*-algebras. Functional calculus.
- C*-algebra fundamentals. Basic definitions. Morphisms between C*-algebras, ideals and quotients. States, representations and the GNS construction.
- The C*-algebra of compact operators. Representations. Applications.
- Unbounded self-adjoint operators and the spectral theorem. Stone-von Neumann theorem.
- The canonical anticommutation relations. Dimensions.
- Group C*-algebras. Crossed product algebras. Amenability. Irrational rotation algebras and other examples.
- Approximation properties. Tensor products, exactness and nuclearity.
- Other topics, as time permits. Cuntz algebras. Extensions. Completely positive mappings.

**Text:**There will be no text, but here are some good online lecture notes for you to refer to (in alphabetical order):

*Bär and Becker - C*-algebras**Klaas Landsman - Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics**Laurent Marcoux - An introduction to operator algebras**Ian Putnam - Lecture notes on C*-algebras**Marc Rieffel - C*-algebras**John Roe - Notes on C*-algebras**Dana Williams - Lecture notes on C*-algebras*

In addition, you may be interested in the (hand-written) lecture notes from the last edition of this course, which was delivered via Zoom. This year’s edition will cover similar material, but not exactly the same material.

Finally, here are some recommended texts (in alphabetical order, again). You shouldn’t have any trouble finding any of them.

***Arveson - An invitation to C*-algebra.*Short and to the point. Excellent on representation theory.*Davidson - C*-algebras by example.*The name says it all.*Dixmier - C*-algebras.*The original text on C*-algebras. Difficult but very valuable; the best source for many points.*Douglas - Banach algebra techniques in operator theory.*A sentimental favorite of mine since I read it as a student. Good on topics related to the Fredholm index.*Fillmore - A user’s guide to operator algebras.*An excellent overview of C*-algebra theory. Sparse on details, though.*Murphy - C*-algebras and operator theory.*The most accessible book in the list, but probably not weighty enough for this course.*Pedersen - C*-algebras and their automorphism groups*. The second standard, go-to text, after Dixmier

**Course and Grading Policies:**Attendance at all lectures is very strongly encouraged! In addition to class attendance and participation, a variety of homework problems and assignments will be provided that I hope will help deepen everyone’s understanding of the material discussed in class. Homework problems and other resources will be made available on the class resource page. There will be no exams. Grades will be assessed on the basis of class participation (50%) and homework (50%).

**Prerequisites:**A good grasp of linear algebra; some familiarity with basic measure theory and basic functional analysis. If in doubt, please contact me.

**Academic Integrity:**Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University's Code of Conduct states that all students should act with personal integrity, respect other students' dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts.

**Students with Disabilities:**Penn State welcomes students with disabilities into the University's educational programs. If you have a disability-related need for reasonable academic adjustments in this course, contact Student Disability Resources at 814-863-1807 (V/TTY). For further information, please visit Student Disability Resources web site: http://equity.psu.edu/student-disability-resources/ . In order to receive consideration for accommodations, you must contact SDR and provide documentation (see the documentation guidelines at the above link). If the documentation supports your request for reasonable accommodations, SDR will provide you with an accommodation letter identifying appropriate academic adjustments. Please share this letter with me and discuss the accommodations with me as soon as possible.

**Information on Available Counseling & Psychological Services:**Students with an interest in obtaining counseling services may wish to contact the Penn State Counseling & Psychological Services Office. More information about the Counseling & Psychological Services Office can be found here: http://studentaffairs.psu.edu/counseling/