Weak Containment and Amenability

This chapter extends the discussion in Chap. 2 to finitely generated groups. We assume throughout this chapter that the Hilbert space \({\mathcal H}\) has infinite dimension. In this case, Kuiper’s theorem [148] asserts that the group of unitaries \(U({\mathcal H})\) is contractible. Recall that given two unitary representations \((\pi , {\mathcal H})\) and \((\rho , {\mathcal H})\) of a group G, we say that \(\pi \) contains \(\rho \) and write \(\rho <\pi \) if \(\rho \) is equivalent to a subrepresentation of \(\pi \) .