C4.10 Operator Algebras (2026-27)
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- Lecturer: Profile: Mehrdad Kalantar
The core prerequistes for this course are the mate-rial found in B4.1 and B4.2 (Functional analysis 1 and 2), including the Hahn Banach theorem; the first half of the A5 topology course (covering compactness in general topological spaces); and the material in C4.1 on quotient Banach spaces, weak*-topologies and the Banach–Alaoglu theo-rem. We imagine most students taking this course would also take C4.1, but it will be possible to take this
course with just the relevant 4-5 lectures from C4.1, which some OMMS students may have seen elsewhere). Spectral theory of operators on Hilbert and Banach spaces will be reviewed at the beginning of the course in the context of Banach algebras, which will make the course accessible to OMMS students who have not seen spectral theory in an earlier course.
The course provides an introduction to the theory of operator algebras, it builds on the two Part B Functional Analysis courses and is complementary to the Part C Further Functional Analysis: Banach Spaces course. The topics covered in this course would provide students with basic knowledge of some modern aspects of operator algebra theory, specially C* - algebras; this equip them with valuable background in preparation for research in many modern analytical research areas (e.g. C* -algebras, von Neumann algebras, noncommutative geometry, analytic and geometric group theory, index theory, random matrix theory and free probability, quantum information theory, mathematical physics, ...). By the end of the course, students will be able to
1. understand the definition of C* -algebras and verify that in concrete examples;
2. understand the GNS construction, and describe concrete GNS rep-resentations;
3. understand the continuous and Borel functional calculus, apply them in concrete problems;
4. describe properties of Fredholm operators in terms of the Calkin al-gebra;
5. understand the basic properties of weak and strong operator topolo-gies, and definition of von Neumann algebras;
6. develop some level of intuition about how operator algebra theory is viewed as noncommutative topology and integration
1. Definition of Banach algebras, and review of spectral theory in this setting [3 Lectures]
2. Definition of C* -algebras, and basic examples [1 Lecture]
3. Commutative C* -algebras and Gelfand’s duality theorem [1 Lecture]
4. Continuous Functional Calculus, and applications [2 Lectures]
5. Positive elements of C* -algebras and positive linear functionals [1 Lecture]
6. The GNS construction [2 Lectures]
7. Ideals, Quotients, Direct products [1 Lecture]
8. Fredholm operators and the Calkin algebra [2 Lectures]
9. Weak and strong operator topologies, Von Neumann algebras, and the bicommutant theorem [2 Lectures]
10. Spectral theorem for normal operators, and Borel functional calculus [1 Lecture]