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. This is a project which might be easy to get started in, but which is a bit speculative. It would obviously involve learning about lots of diverse, but interesting, bits of mathematics.

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