Properties of groups and group algebras

Johnson's Theorem says that a locally compact group G is amenable if and only if the group convolution algebra L^1(G) is amenable as a Banach algebra. Other properties of a group G, such as being compact or discrete, are reflected in various properties of L^1(G).

There are various other properties of G which are defined in terms of properties of the group C*-algebras. For example, being "exact", defined as having the reduced group C*-algebra being exact (in the sense of C*-algebra: in pure algebra, this is being flat as a module).
There are various approximation properties, weaker than being amenable, which can be stated in terms of the group von Neumann algebra. A group has Kazhdan's property (T) if certain properties of its collection of unitary representations holds.

It would be interesting to know how directly such properties can be stated in terms of L^1(G), and if these properties could be extended to a general Banach algebra. Alternatively, can these properties be more easily stated for the Fourier algebra A(G)? Both L^1(G) and A(G) formally capture all the information about G, but extracting this information into a sensible definition can be hard. Once a definition has been found, other Banach algebras could be investigated.

This is a project which should be easy to get started in, but which leads in lots of interesting directions. It would involve learning about diverse, but interesting, bits of mathematics, all revolving around functional analysis.

Analysis group at Leeds

The Analysis group at Leeds is one of the strongest in the UK in abstract analysis. The group comprises 5 members of staff (Vladimir Kisil, Jonathan Partington and Charles Read are also interested in supervising PhD students) together with a research Professor and a number of visiting Fellows. The group currently has six Graduate Students. Researchers at Leeds have interests ranging from the interplay between analysis and algebra (Daws and Read) to links with complex analysis and control theory (Partington) through to applications to Mathematical Physics (Kisil). Leeds hosts a weekly seminar series (which has close links with York University) and is currently the organising node for the North British Functional Analysis Seminar series.