| Syllabus |
Lecture Notes (Updated April 27, 2020) |
Videos of the lectures |
Jesse Peterson's homepage |
NCGOA Spring Institute 2020 |
| Note: Due to concerns over the coronavirus pandemic all lectures will be taught online beginning March 16th. The lectures will take place 2:10-3:00pm each lecture day. The lectures will stream at: https://vanderbilt.zoom.us/s/606291971 |
| Date | Material covered | Video | HW problems and assignments | Remarks |
| Jan 6 |
Banach Algebras
Convolution algebras Spectrum of an operator The Gelfand-Mazur theorem |
Lecture 1 | Homework 1 ( tex) | Homework is due Friday, January 10, by 4:00pm |
| Jan 8 |
Gelfand's spectral radius formula
C*-algebras |
Lecture 2 | The second half of the lecture did not record properly. | |
| Jan 10 | Gelfand transform | Lecture 3 | ||
| Jan 13 | Continuous functional calculus | Lecture 4 | ||
| Jan 15 | Banach and C*-algebras without units. | Lecture 5 | Homework 2 ( tex) | Homework is due Wednesday, January 22, by 4:00pm |
| Jan 17 | The positive cone | Lecture 6 | ||
| Jan 22 | Approximate identities | Lecture 7 | ||
| Jan 24 |
States
The Gelfand-Naimark-Segal construction |
Lecture 8 | ||
| Jan 27 |
Trace class operators
Hilbert-Schmidt operators Compact operators The Fredholm alternative Operator topologies Von Neumann's bicommutant theorem The Kaplansky density theorem The spectral theorem Borel functional calculus spectral measures |
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| Jan 29 | Abelian von Neumann algebras | |||
| Jan 31 | Standard probability space | |||
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| Mar 11 | This lecture has been postponed. | |||
| Mar 13 | This lecture has been postponed. | |||
| Mar 16 | Centrally orthogonal projections
The Comparison Theorem for projections Types of projections |
Lecture | Slides | |
| Mar 18 | finite, semi-finite, and purely infinite projections | Lecture | Slides | |
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