Topics in operator algebras
This course extends the introductory study of $C^*$-algebras and $K$-theory I and II introduced last academic year. It starts with supplementary explanation and exposition to discussions conducted last semester, including basic facts on hereditary $C^*$-subalgebras, extension of states and holomorphic functional calculus. The next subject is Sakai's approach to the theory of von Neumann algebras.
Lecturer
Date
1st September ~ 30th November, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Thursday | 14:20 - 16:05 | A3-1-301 | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Algebra; Complex Analysis; Basics on $C^*$-algebras; Measure Theory; Real Analysis
Syllabus
1. Gelfand duality.
2. Hereditary C∗-subalgebras.
3. Extension of states on C∗-algebras.
4. Maximal and spatial norms for C∗-tensor products.
5. Holomorphic functional calculus.
6. Extreme points of closed unit balls of C∗-algebras.
7. Topological vector spaces; dual pairs.
8. Locally convex topological vector spaces; seminorms.
9. Polars, Mackey topology and the Mackey-Arens theorem.
10. An introduction to topological tensor product.
11. Topologies on bounded linear operators on a Hilbert space.
12. Trace-class operators and associated dual pairs.
13. Decomposable measure spaces, Radon-Nikodym type theorem and associated dual pairs.
14. Sakai’s approach to von Neumann algebras; basics.
15. Topologies on von Neumann algebras, connections and correspondence with those on bounded linear operators.
16. Ideals of von Neumann algebras, supports of states.
17. Gelfand-Naimark type theorem for von Neumann algebras.
18. Representations of von Neumann algebras; a proposition in last semester.
19. GNS construction for von Neumann algebras.
2. Hereditary C∗-subalgebras.
3. Extension of states on C∗-algebras.
4. Maximal and spatial norms for C∗-tensor products.
5. Holomorphic functional calculus.
6. Extreme points of closed unit balls of C∗-algebras.
7. Topological vector spaces; dual pairs.
8. Locally convex topological vector spaces; seminorms.
9. Polars, Mackey topology and the Mackey-Arens theorem.
10. An introduction to topological tensor product.
11. Topologies on bounded linear operators on a Hilbert space.
12. Trace-class operators and associated dual pairs.
13. Decomposable measure spaces, Radon-Nikodym type theorem and associated dual pairs.
14. Sakai’s approach to von Neumann algebras; basics.
15. Topologies on von Neumann algebras, connections and correspondence with those on bounded linear operators.
16. Ideals of von Neumann algebras, supports of states.
17. Gelfand-Naimark type theorem for von Neumann algebras.
18. Representations of von Neumann algebras; a proposition in last semester.
19. GNS construction for von Neumann algebras.
Reference
1. Nathanial P. Brown and Narutaka Ozawa, C∗-algebras and finite- dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387 (2009h:46101)
2. Shôichirô Sakai, C∗-algebras and W∗-algebras, Classics in Mathematics, Springer-Verlag, Berlin, 1998, Reprint of the 1971 edition. MR 1490835 (98k:46085)
2. Shôichirô Sakai, C∗-algebras and W∗-algebras, Classics in Mathematics, Springer-Verlag, Berlin, 1998, Reprint of the 1971 edition. MR 1490835 (98k:46085)
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
He graduated from the University of Tokyo in 2018, and then became an associate research fellow at University of Tokyo. After finishing his postdocteral position at East China Normal University in 2022, He joined BIMSA as an assistant professor in 2023. He's recent research interests lie in classification theory of C*-algebras, C*-dynamical systems and topological dynamical systems.