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 | |||
Feb 3 | ||||
Feb 5 | ||||
Feb 7 | ||||
Feb 10 | ||||
Feb 12 | ||||
Feb 14 | ||||
Feb 17 | ||||
Feb 19 | ||||
Feb 21 | ||||
Feb 24 | ||||
Feb 26 | ||||
Feb 28 | ||||
Mar 9 | ||||
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 | |
Mar 20 | ||||
Mar 23 | ||||
Mar 25 | ||||
Mar 27 | ||||
Mar 30 | ||||
Apr 1 | ||||
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Apr 6 | ||||
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Apr 10 | ||||
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Apr 15 | ||||
Apr 17 | ||||
Apr 20 |