# C* Algebras and Spectral Theorem

- Institution: IISc Bangalore
- Department: Mathematics
- Teacher: Prof. E K Narayanan
- Level: Graduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111108164)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111108164)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

Aim is to prove the spectral theorem for bounded normal operators using C* algebras and projection valued measures. After review of spectral theorem for compact normal operators we will introduce Banach algebras, C* algebras and prove Gelfand-Naimark theorem. Spectral theorem will be proved after a detailed study of projection valued measures and their properties

Intended audience: MSc 2nd year , PhD students and college lecturers

Prerequisites: First course in functional analysis is a must. See http://math.iisc.ac.in/all-courses/ma223.html

## Lectures

1. [Finite dimensional Spectral theorem](https://invisiblenetwork.ai/learn/nptel/111108164/33cb5e638677) (YouTube)
2. [Compact operators](https://invisiblenetwork.ai/learn/nptel/111108164/2c6a066937d7) (YouTube)
3. [Spectral theorem for Compact self-adjoint operators](https://invisiblenetwork.ai/learn/nptel/111108164/784909c2d9d0) (YouTube)
4. [Spectral theorem for compact Normal operators](https://invisiblenetwork.ai/learn/nptel/111108164/379e2bf5f4eb) (YouTube)
5. [Banach algebras](https://invisiblenetwork.ai/learn/nptel/111108164/ab3fe45b4063) (YouTube)
6. [Gelfand-Mazur theorem](https://invisiblenetwork.ai/learn/nptel/111108164/fe8820e09012) (YouTube)
7. [Spectral radius](https://invisiblenetwork.ai/learn/nptel/111108164/073ea261416a) (YouTube)
8. [Multiplicative functionals](https://invisiblenetwork.ai/learn/nptel/111108164/8d29d8117462) (YouTube)
9. [Gelfand transform-I](https://invisiblenetwork.ai/learn/nptel/111108164/66a5a212f944) (YouTube)
10. [Gelfand transform -II](https://invisiblenetwork.ai/learn/nptel/111108164/8439cfed3fa4) (YouTube)
11. [C* algebras](https://invisiblenetwork.ai/learn/nptel/111108164/c43f6cd945af) (YouTube)
12. [Examples and Wiener’s theorem](https://invisiblenetwork.ai/learn/nptel/111108164/e287d252892b) (YouTube)
13. [Gelfand-Naimark theorem](https://invisiblenetwork.ai/learn/nptel/111108164/e8fce736ae18) (YouTube)
14. [Non-unital Banach algebras](https://invisiblenetwork.ai/learn/nptel/111108164/bcc9cde36e95) (YouTube)
15. [Non-unital C* algebra](https://invisiblenetwork.ai/learn/nptel/111108164/3334e29ad1ca) (YouTube)
16. [Gelfand transform of non-unital C*algebras](https://invisiblenetwork.ai/learn/nptel/111108164/b74d0b768dcc) (YouTube)
17. [Gelfand-Naimark theorem for non-unital C* algebras](https://invisiblenetwork.ai/learn/nptel/111108164/ef645e2d72f1) (YouTube)
18. [Continuous functional calculus](https://invisiblenetwork.ai/learn/nptel/111108164/1cff441e25bc) (YouTube)
19. [Bounded functional calculus -I](https://invisiblenetwork.ai/learn/nptel/111108164/fcdfbf55c457) (YouTube)
20. [Bounded functional calculus -II](https://invisiblenetwork.ai/learn/nptel/111108164/3ac7a449ea96) (YouTube)
21. [Projection valued measures](https://invisiblenetwork.ai/learn/nptel/111108164/ac8a45cc0ef2) (YouTube)
22. [Bounded functional calculus with respect to a projection valued measure](https://invisiblenetwork.ai/learn/nptel/111108164/15551d961340) (YouTube)
23. [Spectral Theorem I](https://invisiblenetwork.ai/learn/nptel/111108164/43cd21be1849) (YouTube)
24. [Spectral theorem II](https://invisiblenetwork.ai/learn/nptel/111108164/c4fe194f4bbd) (YouTube)
25. [Some applications](https://invisiblenetwork.ai/learn/nptel/111108164/8568704d0edc) (YouTube)
26. [Spectral theorem for a bounded normal operator](https://invisiblenetwork.ai/learn/nptel/111108164/e2b1d3551dbc) (YouTube)
27. [Resolution of identity -I](https://invisiblenetwork.ai/learn/nptel/111108164/dcbf6cc107a5) (YouTube)
28. [Resolution of identity -II](https://invisiblenetwork.ai/learn/nptel/111108164/1d4043da99d5) (YouTube)
29. [Resolution of identity-III](https://invisiblenetwork.ai/learn/nptel/111108164/db661cbfc3f7) (YouTube)
30. [Resolution of identity-IV](https://invisiblenetwork.ai/learn/nptel/111108164/2bf7757f1d31) (YouTube)
31. [Equivalence of various forms of spectral theorems -I](https://invisiblenetwork.ai/learn/nptel/111108164/1a7a5ca3bc58) (YouTube)
32. [Equivalence of various forms of spectral theorems -II](https://invisiblenetwork.ai/learn/nptel/111108164/ee55abdb7ff5) (YouTube)
33. [Spectrum of a self-adjoint operator-I](https://invisiblenetwork.ai/learn/nptel/111108164/df4238f6ef31) (YouTube)
34. [Spectrum of a self-adjoint operator-II](https://invisiblenetwork.ai/learn/nptel/111108164/90d21a9686c5) (YouTube)
35. [Commuting family of self-adjoint operators](https://invisiblenetwork.ai/learn/nptel/111108164/32b299443ad2) (YouTube)
36. [Continuous functional calculus for commuting family of self-adjoint operators -I](https://invisiblenetwork.ai/learn/nptel/111108164/0fa459964c58) (YouTube)
37. [Continuous functional calculus for commuting family of self-adjoint operators -II](https://invisiblenetwork.ai/learn/nptel/111108164/1321f832baec) (YouTube)
38. [Fuglede’s theorem](https://invisiblenetwork.ai/learn/nptel/111108164/f5c7373d7fad) (YouTube)
39. [Spectral theorem for commuting finite family of normal operators](https://invisiblenetwork.ai/learn/nptel/111108164/c6b8af88c7bc) (YouTube)
40. [Multiplicity theory](https://invisiblenetwork.ai/learn/nptel/111108164/eed374056946) (YouTube)