Research in Groups: *-Representation Theory of *-Algebras

Winter term 2016/2017

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This Research in Groups aims at master students in the programs International Master in Mathematics, in Mathematics, and Mathematical Physics. It may also be interesting for students in the master program Physics.

In mathematical models of physical systems, the physical observables are typically described by C*-algebras or von Neumann algebras. While this gives a very elegant and powerful spectral calculus, many situations will not directly yield a such nice classes of algebras. In various quantization theories the construction of C*-algebras is difficult or unclear. One way out is to focus on the algebraic features before taking into account the analytic issues as well. This is the main motivation for considering *-algebras without any analysis involved. Beyond quantization theories, other important examples are group algebras or universal enveloping algebras of Lie algebras but also algebras of differential operators. Here one typically has by no means a C*-norm available.

The aim of this RiG is to find a common algebraic framework for a reasonable representation theory of such algebras. It turns out that aspects of positivity can be formulated in a entirely algebraic way yielding interesting structures for the representation theory. In general, it will be difficult if not impossible to understand the representation theory of a given algebra completely. However, and this is quite surprising, it might be possible to compare it to the representation theory of a different algebra and determine whether or not the two algebras have the same representation theory. This is the main task of Morita theory, which we will present both in a purely ring-theoretic context and in an adapted version for *-algebras. We will develop the necessary category-theoretic notions to put the question of Morita equivalence in the right perspective.

The course consists of essentially two components: first, a lecture where the basic notions of representation theory are explained. The preliminary program for the lecture component includes the following topics:

The second component will be a seminar by the students on more particular topics. We expect the participants to write a small proceeding-like summary of their seminar talks.


For this RiG you will not need much prerequisites. In fact, a good knowledge in (multi-) linear algebra will be sufficient for most things. Depending on the examples you want to discuss, some ideas about Lie algebras, differential geometry, algebra, functional analysis might be useful but certainly not necessary.

However, if you are in doubt, please contact me directly and we will find a solution. I will explain the necessary things either in the lecture part or directly.


The following list of references will be discussed in the first meeting. More references will be given individually.


If there are collisions with other lectures or seminars, please contact me early: maybe one can still shift things around a bit.


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