Analysis and Probability Seminar

The existence of a Kazhdan projection in the maximal group C*-algebra characterises Kazhdan's property (T). For a given unitary representation of a property (T) group, this projection can be viewed as the projection onto the 0 th-degree cohomology group with coefficients in the universal representation. In joint work with K. Li and P. Nowak, we employed higher degree cohomology groups with more general coefficients to define a generalisation of Kazhdan projections. These are called higher Kazhdan projections. In this talk, I will introduce this generalisation and provide concrete examples of higher Kazhdan projections. In contrast to the classical setting, higher Kazhdan projections do not need to come from a property (T) group. Moreover, the K-theory classes of these projections can be non-zero in the K-theory of the reduced group C*-algebra. I will discuss joint work with H. Wang, which describes these phenomena in the case of free products Z_2 * Z_3. Time permitting, I will discuss applications to various L²-Betti numbers which employ higher Kazhdan projections.