Operator algebras

and their application in physics.

Lecturer: prof. dr hab. Jan Dereziński, Department of Mathematical Methods in Physics, Warsaw University,

visiting Erwin Schroedinger Institute in February-May 2005 as a senior research fellow

Email: jan.derezinski@fuw.edu.pl

Place: ESI, lecture hall

Time: March 2, 3, 9, 10, April 6,13,20,27 and May 4, 11, 18, 25 at 10:15 -12:00, ESI lecture hall

Seminar: to be announced

Period of the course: March, April, May 2005

Course description The course will be devoted to selected topics about C* and W* algebras. Special attention will be devoted to their applications in quantum statistical physics. KMS states, which are used to describe thermal equilibrium states of quantum systems, will be discussed. Concrete physical examples will be presented. No background in theory of operator algebras will be needed.

List of topics

1. Basic definitions concerning C* and W* algebras
2. States and representations
3. Examples motivated by physics
   • Quasi-local algebras.
   • Free Bose gas.
   • Free Fermi gas.
4. KMS states and the Tomita-Takesaki theory.
5. Positive cone and the standard representation.

• Bratteli D., Robinson, D.: ``Operator Algebras and Quantum Statistical Physics'', vol. I and II
• Takesaki, M.: ``Theory of Operator Algebras'', vol. I and II
• Derezinski, J. ``Fermi Golden Rule and open quantum systems'', lecture notes of a minicourse given at Summer School "Open Quantum Systems", Grenoble 2003
• Derezinski, J.: ``Representations of canonical commutation and anticommutation relations'', lecture notes of a minicourse given at Summer School "Large Coulomb Systems", Nordfjordeid 2003 ( PostScript file, version of Jan. 2005)
  

(Further lecture notes will be available later on)