MSc Project „U*-Algebras“
The study of operator algebras originated in quantum physics, but it is also interesting for purely mathematical reasons. It has led to many results about the structure and properties of von Neumann algebras and C*-algebras. This project is about U* algebras, a slightly more general type of complex *-algebra all of whose elements can be written as finite linear combinations of unitary elements. U*-algebras have found some applications in mathematical physics, but their mathematical theory has received very little attention and is not well developed yet.
The goal of this project is to develop the basic theory of U*-algebras. The class of U* algebras includes all C*-algebras and certain dense sub-algebras thereof. Many results about C*-algebras are expected to generalise to U*-algebras without difficulty (e.g. the use of states to construct GNS-representations and the uniqueness of a norm with suitable properties). Some ideas might work better for U*-algebras than for C*-algebras (e.g. the tensor product of two U*-algebras should be another U*-algebra, but the corresponding statement for C*-algebras fails). If time permits the project can also address more challenging questions, e.g. whether all U*-algebras are dense sub-algebras of a C*-algebra.
This project is suitable for students who are interested in analysis and who like to work with abstract algebraic structures.
- D. E. Evans and J. T. Lewis,
„Dilations of irreversible evolutions in algebraic quantum theory,“
Vol. 24 of Communications of DIAS Series A, Dublin (1977).
- E. Binz, R. Honegger and A. Rieckers,
„Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space,“
J. Math. Phys. 45, 2885-2907 (2004)
About the supervisor:
This project will be supervised by Prof. Ko Sanders who is working at DCU Dublin and will join us at FAU for one year starting in fall 2022. A fairly recent overview of his research interests and scientific profile can be found here.
This project should start in fall 2022. If you are interested, please get in touch via email to <firstname.lastname@example.org>.