MA8107 Operator Algebras, fall 2024

Lecturer: Eduard Ortega

We will cover the basic theory on C*-algebras, prominent examples like group C*-algebras and a brief introduction to Quantum Information Theory.

The goal is to expose students to the basic notions in the theory of operator algebras and to prepare them to understand fundamentals used in current areas of active research.

In order to gain a deeper understanding of the abstract concepts we examine a number of examples that illustrate different aspects of the theory, such as approximately finite algebras, group algebras, and convolution algebras. Finally we will introduce the basic concepts and definitions on Quantum Information theory like Quantum systems, Quantum channels and completely positive maps.

Among the topics to be covered are the following:

• Banach Algebra basics, Commutative C*-algebras, positive elements, representations of C*-algebras.
• C*-algebra of compact operators.
• AF C*-algebras
• K-theory for C*-algebras.
• Group C*-algebras, amenable groups, the free group.
• Quantum systems, quantum channels.
• Completely positive maps and Stinespring's theorem.

First day: Tuesday 20.08

* Tuesday 10:15 - 12:00 in 656 (SB2) * Thursday 12:15 - 14:00 in 656 (SB2)

The evaluation of the course will be done through an oral exam on the basic topics of the course, plus the presentation of one project.

There is also the option to write a project on a variety of topics and in various degrees of complexity and sizes. Potential topics will be announced during the course.

Functional analysis TMA4230 is the official prerequisite, but Linear Methods should suffice for an understanding of a large part of the material.

• “C*-algebras and operator theory” by Gerard J. Murphy. Boston : Academic Press, 1990.
• ”C*-algebras by example” av Kenneth R. Davidson. Fields Institute monographs; vol. 6 Providence, R.I. : American Mathematical Society, 1996.
• "Morita equivalence and continuous-trace C*-algebras" by Iain Raeburn, Dana P. Williams. Mathematical surveys and monographs ; no. 60. Providence, R.I. : American Mathematical Society, 1998.
• "An Introduction to Noncommutative Geometry" by Joseph C. Várilly. EMS series of lectures in mathematics European Mathematical Society, 2006.
• "Quantum information, and introduction" by M. Hayashi. Springer.

You can follow the books in the bibliography or this notes from Ian Putnam, and Quantum information from Vern Paulsen.

Every week we are going to update the course lecture notes. notes_ma8107.pdf