Operator Theory

This page is devoted to the Fall 2025 course MATH4160/5160 at Dalhousie University.

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

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

There will be a midterm in October. Students in Math 5160 have to give a 10-20 minute presentation in November.

There will be a total number of 6 homework assignments. The exercise sheets will appear here. The first assignment is due on September 18.

Here is a rough outline for the course: Lecture 1 September 2: Introduction lecture
Lecture 2 September 4: Operators on Hilbert spaces
Lecture 3 September 9: The category of C*-algebras and convergent power series
Lecture 4 September 11: Holomorphic calculus and integration 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 DUE
Lecture 7 September 23: The spectrum of a commutative C*-algebra
Lecture 8 September 25: Gelfand duality ASSIGNMENT 2 DUE
September 30: holiday
Lecture 9 October 2: The GNS construction
Lecture 10 October 7: Topologies on B(H) and related spaces
Lecture 11 October 9: Introduction to von Neumann algebras ASSIGNMENT 3 DUE
Lecture 12 October 14: Projections in von Neumann algebras
Lecture 13 October 16: Carriers and commutants
MIDTERM (material until October 7)
Lecture 14 October 21: Polar decomposition LECTURE WILL BE ONLINE
Lecture 15 October 23: The double commutant theorem LECTURE WILL BE ONLINE
Lecture 16 October 28: Stone(an) spaces, Boolean algebras and the Pierce spectrum
Lecture 17 October 30: Review of measure theory ASSIGNMENT 4 DUE
Lecture 18 November 4: Lp-spaces and the von Neumann algebra Linfty(X)
Lecture 19 November 6: Measure-theoretic Gelfand duality ASSIGNMENT 5 DUE
November 11: reading week
November 13: reading week
Lecture 20 November 18: The von Neumann algebra completion of a C*-algebra and the measure space associated to a topological space
Lecture 21 November 20: LECTURE WILL BE ONLINE Traces on von Neumann algebras and the relationship with measures ASSIGNMENT 6 DUE
Lecture 22 November 25: The predual of a von Neumann algebra
Lecture 23 November 27: Factors, types and the classification of von Neumann algebras