MATH 582, Spring 2026
Course Title: Introduction to C*-algebra theory
Instructor: Nigel Higson, 228 McAllister Building
Class Meetings: Tuesdays and Thursdays, 12:05-1:20, in 209 Thomas. Note the new room from 20 January onwards!
Office Hours: Wednesdays 2:00-4:00 (starting January 21) or by appointment. Contact me in class or send me an email. My PSU ID is ndh2.
Content, Goals, Objectives, Expectations: 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, and give examples and applications, roughly as follows:
I hope that by the end of the class participants will be aware of some of the ways in which C*-algebra theory can contribute useful tools and perspectives to a variety of topics in mathematics and physics. For those who wish to reach this goals, regular attendance of classes is very strongly encouraged, and steady work on problems, beyond the required homework problems, will be essential.
Text: There will be no text, but here are some good online lecture notes for you to refer to (in alphabetical order):
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.
Class page: I will keep a record of events in class, and post homework problems and various supplementary materials here.
Homework: A variety of homework problems will be provided that I hope they will help deepen your understanding of the material discussed in class. I will require you to hand in solutions to a minority of the homework problems.
Exams: There will be no exams.
Grading Policy: Grades will be assessed on the basis of class participation (25%) and homework (75%).
Prerequisites: A good grasp of linear algebra; some familiarity with basic measure theory and basic functional analysis. If in doubt, please contact me.
Contacting you: If I need to contact you in a hurry (because of a class cancellation, for instance) I will do so by email, to your psu.edu addresses. Please monitor your email!
Changes to the syllabus: This syllabus is subject to change. If any change is made, then I will notify registered class participants of it by email.
And now, the boilerplate …
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.
According to Penn State policy G-9: Academic Integrity, an academic integrity violation is “an intentional, unintentional, or attempted violation of course or assessment policies to gain an academic advantage or to advantage or disadvantage another student academically.” Unless your instructor tells you otherwise, you must complete all course work entirely on your own, using only sources that have been permitted by your instructor, and you may not assist other students with papers, quizzes, exams, or other assessments. If your instructor allows you to use ideas, images, or word phrases created by another person (e.g., from Course Hero or Chegg) or by generative technology, such as ChatGPT, you must identify their source. You may not submit false or fabricated information, use the same academic work for credit in multiple courses, or share instructional content. Students with questions about academic integrity should ask their instructor before submitting work.
Students facing allegations of academic misconduct may not drop/withdraw from the affected course unless they are cleared of wrongdoing (see G-9: Academic Integrity). Attempted drops will be prevented or reversed, and students will be expected to complete course work and meet course deadlines. Students who are found responsible for academic integrity violations face academic outcomes, which can be severe, and put themselves at jeopardy for other outcomes which may include ineligibility for Dean’s List, pass/fail elections, and grade forgiveness. Students may also face consequences from their home/major program and/or The Schreyer Honors College.
Education Equity and Reporting of Bias: Consistent with University Policy AD29, students who believe they have experienced or observed a hate crime, an act of intolerance, discrimination, or harassment that occurs at Penn State are urged to report these incidents as outlined on the University’s Report Bias webpage.
Disability Accommodation: Penn State welcomes students with disabilities into the University’s educational programs. Every Penn State campus has an office for students with disabilities. Student Disability Resources (SDR) website provides contact information for every Penn State campus. For further information, please visit the Student Disability Resources website.
In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: See documentation guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations request with them as early as possible. You must follow this process for every semester that you request accommodations.
Students with disabilities participating in internship, practicum, student teaching, or other experiential learning opportunities as part of their degree requirements may also be eligible for reasonable accommodations to ensure equal access and opportunity. These accommodations are determined through an interactive process involving the student, their University supervisor, and the site supervisor. Student Disability Resources can assist students with identifying potential barriers, facilitating accommodation requests, and coordinating with University supervisors to promote inclusive learning experiences.
Counseling and Psychological Services: Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional wellbeing. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of clients’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation.
Counseling and Psychological Services at University Park (CAPS): 814-863-0395
Penn State Crisis Line (24 hours/7 days/week): 877-229-6400
Crisis Text Line (24 hours/7 days/week): Text LIONS to 741741
Instructor: Nigel Higson, 228 McAllister Building
Class Meetings: Tuesdays and Thursdays, 12:05-1:20, in 209 Thomas. Note the new room from 20 January onwards!
Office Hours: Wednesdays 2:00-4:00 (starting January 21) or by appointment. Contact me in class or send me an email. My PSU ID is ndh2.
Content, Goals, Objectives, Expectations: 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, and give examples and applications, 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. Sturm-Liouville theory, after Weyl. Stone-von Neumann theorem.
- The canonical anticommutation relations. Dimensions.
- Group C*-algebras. Crossed product algebras. Amenability. Irrational rotation algebras and other examples.
- Other topics, as time permits. Cuntz algebras. Extensions. Completely positive mappings.
I hope that by the end of the class participants will be aware of some of the ways in which C*-algebra theory can contribute useful tools and perspectives to a variety of topics in mathematics and physics. For those who wish to reach this goals, regular attendance of classes is very strongly encouraged, and steady work on problems, beyond the required homework problems, will be essential.
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
Class page: I will keep a record of events in class, and post homework problems and various supplementary materials here.
Homework: A variety of homework problems will be provided that I hope they will help deepen your understanding of the material discussed in class. I will require you to hand in solutions to a minority of the homework problems.
Exams: There will be no exams.
Grading Policy: Grades will be assessed on the basis of class participation (25%) and homework (75%).
Prerequisites: A good grasp of linear algebra; some familiarity with basic measure theory and basic functional analysis. If in doubt, please contact me.
Contacting you: If I need to contact you in a hurry (because of a class cancellation, for instance) I will do so by email, to your psu.edu addresses. Please monitor your email!
Changes to the syllabus: This syllabus is subject to change. If any change is made, then I will notify registered class participants of it by email.
And now, the boilerplate …
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.
According to Penn State policy G-9: Academic Integrity, an academic integrity violation is “an intentional, unintentional, or attempted violation of course or assessment policies to gain an academic advantage or to advantage or disadvantage another student academically.” Unless your instructor tells you otherwise, you must complete all course work entirely on your own, using only sources that have been permitted by your instructor, and you may not assist other students with papers, quizzes, exams, or other assessments. If your instructor allows you to use ideas, images, or word phrases created by another person (e.g., from Course Hero or Chegg) or by generative technology, such as ChatGPT, you must identify their source. You may not submit false or fabricated information, use the same academic work for credit in multiple courses, or share instructional content. Students with questions about academic integrity should ask their instructor before submitting work.
Students facing allegations of academic misconduct may not drop/withdraw from the affected course unless they are cleared of wrongdoing (see G-9: Academic Integrity). Attempted drops will be prevented or reversed, and students will be expected to complete course work and meet course deadlines. Students who are found responsible for academic integrity violations face academic outcomes, which can be severe, and put themselves at jeopardy for other outcomes which may include ineligibility for Dean’s List, pass/fail elections, and grade forgiveness. Students may also face consequences from their home/major program and/or The Schreyer Honors College.
Education Equity and Reporting of Bias: Consistent with University Policy AD29, students who believe they have experienced or observed a hate crime, an act of intolerance, discrimination, or harassment that occurs at Penn State are urged to report these incidents as outlined on the University’s Report Bias webpage.
Disability Accommodation: Penn State welcomes students with disabilities into the University’s educational programs. Every Penn State campus has an office for students with disabilities. Student Disability Resources (SDR) website provides contact information for every Penn State campus. For further information, please visit the Student Disability Resources website.
In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: See documentation guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations request with them as early as possible. You must follow this process for every semester that you request accommodations.
Students with disabilities participating in internship, practicum, student teaching, or other experiential learning opportunities as part of their degree requirements may also be eligible for reasonable accommodations to ensure equal access and opportunity. These accommodations are determined through an interactive process involving the student, their University supervisor, and the site supervisor. Student Disability Resources can assist students with identifying potential barriers, facilitating accommodation requests, and coordinating with University supervisors to promote inclusive learning experiences.
Counseling and Psychological Services: Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional wellbeing. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of clients’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation.
Counseling and Psychological Services at University Park (CAPS): 814-863-0395
Penn State Crisis Line (24 hours/7 days/week): 877-229-6400
Crisis Text Line (24 hours/7 days/week): Text LIONS to 741741