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



Schedule:
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
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
Apr 3
Apr 6
Apr 8
Apr 10
Apr 13
Apr 15
Apr 17
Apr 20