Math 8120 - Operator algebras - Vanderbilt University - Spring 2020

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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