Property \((T)\) for \(II_ 1\) factors and unitary representations of Kazhdan groups (Q1318073)

A discrete group \(\Gamma\) has Kazhdan's property (T) if and only if pointwise convergence to 1 of a sequence of positive definite functions on \(\Gamma\) implies uniform convergence. This condition is used to give a new proof of the result of Connes and Jones that property (T) for an ICC group is equivalent to property (T) for the corresponding group factor. Let \(G\) be a discrete group such that \(C_ 0(G)\) has an approximate unit consisting of positive definite functions; for example the free group on two generators. Approximation by compact completely positive maps shows that if a Kazhdan group \(\Gamma\) has a faithful unitary representation in the group von Neumann algebra of \(G\) then \(\Gamma\) is maximally almost periodic. Some criteria are given for \(II_ 1\) factors to have property (T).