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

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.

Project 1 CAR algebras: Define the algebra defined by Canonical Anti-commutative Relations. See that the CAR algebra is an AF-algebra. You can look for example this notes of Christopher Hawthorne.

Project 2 Classification of UHF-algebras. A UHF-algebra is a special class of AF-algebra. You should see that UHF-algebras are simple and that can be classified by its associated supernatural number. You can look for example this notes of Christopher Hawthorne.

Project 3 The non-commutative torus embeds into an AF-algebra. There are two papers one can follows: Paper 1 or/and Paper 2

Project 4 The graph C*-algebra. In this projects one should learn how to given a directed graph to define a C*-algebra. Follow the notes from Iain Raeburn.

Project 5 The group algebras of the free group. See that that reduced group algebra of the free group with 2 generators is not isomorphic to its full group C*-algebra. You can see it for example in Ian Putnam's notes

Project 6 Amenable groups. Amenable groups are those such that the reduced and the full group C*-algebras coincide. Study the basic properties and different charactarizations of amenable groups. You can follow the first chapters of Juschenko book

Project 7 Swan Theorem. Swan theorem relates the equivalence classes of vector bundles of a topological space X with the projective modules of C(X). Swan Theorem

Project 8 Quantum Harmomic analysis. One can extend the definitions of Harmonic analysis to operators. This provide a nice decomposition of operators in terms of "non-commutative" wave functions. This has several application to fields like Gabor analysis and quantum theory. You can read the original paper from Werner or Skreittingland thesis

Project 9 Quantum circuits (The Quantum Fourier Transform) One of the powers of Quantum theory is to do computations. Digital circuits for classical computation are made from logic gates acting on Boolean variables. Quantum circuits are the quantum counterparts of digital circuits. Quantum Circuits