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.
讲师
日期
2025年09月01日 至 11月30日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周四 | 14:20 - 16:05 | A3-1-301 | ZOOM 06 | 537 192 5549 | BIMSA |
修课要求
Algebra; Complex Analysis; Basics on $C^*$-algebras; Measure Theory; Real Analysis
课程大纲
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.
参考资料
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)
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
视频公开
不公开
笔记公开
不公开
语言
英文
讲师介绍
贺卓丰2018年毕业于东京大学，后成为该校副研究员。2022年华东师范大学博士后出站，2023年加入北京雁栖湖应用数学研究院任助理研究员。现在的研究兴趣包括C*代数的分类理论、C*动力系统和拓扑动力系统。