Operator Theory

I taught MATH4160/5160 at Dalhousie University in Fall 2025.

Student feedback

“The course was both challenging and useful. It not only teaches you about extending the toolbox of linear algebra to infinite dimensions but highlights some of the most important cases of algebraic geometric duality. Luuk was able to pull together many different threads of reasoning into what felt like a course which was cohesive and comprehensive for the time allotted in the semester. Would recommend to any mathematics undergrad especially mathematical physics.”

Lectures took place Tuesdays 11:35-12:55 and Thursdays 11:35-12:55 in Chase 227. Office hours were Wednesdays 14:35-15:25 in Chase 224. There were no tutorials.

The main reference material for the course was the PhD thesis “The Category of Von Neumann Algebras” by Bram Westerbaan.

There was a midterm in October. Students in Math 5160 were required to give a 10-20 minute presentation on November 25.

There were 6 homework assignments in total. The exercise sheets were posted on this page. The first assignment was due on September 18.

The course followed this outline:

Lecture 1 September 2: Introduction lecture
Lecture 2 September 4: Operators on Hilbert spaces
Lecture 3 September 9: The category of C*-algebras
Lecture 4 September 11: Convergent power series and holomorphic functions in Banach algebras
Lecture 5 September 16: The spectrum of an element of a C*-algebra
Lecture 6 September 18: Positive elements in a C*-algebra ASSIGNMENT 1 WAS DUE
Lecture 7 September 23: The spectrum of a commutative C*-algebra
Lecture 8 September 25: The Gelfand transform ASSIGNMENT 2 WAS DUE
September 30: holiday
Lecture 9 October 2: Gelfand duality I
Lecture 10 October 7: Gelfand duality II
Lecture 11 October 9: Review lecture ASSIGNMENT 3 WAS DUE
October 14: MIDTERM (material until October 7)
Lecture 12 October 16: Projections in C*-algebras
Lecture 13 October 21: Topologies on B(H) and commutants LECTURE WAS ONLINE
Lecture 14 October 23: The double commutant theorem LECTURE WAS ONLINE
Lecture 15 October 28: Trace class operators
Lecture 16 October 30: The predual of a von Neumann algebra and normal homomorphisms ASSIGNMENT 4 WAS DUE
Lecture 17 November 4: Continuous functional calculus Lecture 18 November 6: Stone(an) spaces, Boolean algebras and the Pierce spectrum ASSIGNMENT 5 WAS DUE
November 11: reading week
November 13: reading week
Lecture 19 November 18: Review of measure theory and the von Neumann algebra Linfty(X)
Lecture 20 November 20: Measure-theoretic Gelfand duality LECTURE WAS ONLINE
Lecture 21 November 25: Student presentations ASSIGNMENT 6 WAS DUE
Lecture 22 November 27: Conclusion and review lecture
December 15: FINAL EXAM (8:30AM-11:30AM)