Let H be a separable infinite-dimensional Hilbert space. Denote U(H) the unitary group of H, provided with the operator norm topology. By a result of Kadison, U(H) has a, up to complex scalars, unique closed normal subgroup \(U(H)_{\infty}\) consisting of unitary operators U of the form \(U=I+T\) with T a compact operator. Kirillov and Ol'shankij have made a detailed study of the theory of unitary (strong operator continuous) representations of \(U(H)_{\infty}\). Motivated by some questions from gauge theory, the author tries to link the representation theory of U(H) and \(U(H)_{\infty}\). His main result is: every unitary representation of \(U(H)_{\infty}\) on a separable Hilbert space has a unique extension to U(H). The proof uses the analogue of the following fact about the Calkin algebra \(L(H)/L(H)_{\infty}:\) if \(\pi\) is a non-trivial *- representation of the Calkin algebra and \(T\in L(H)\) is a normal operator, then \(\lambda\) is an eigenvalue for \(\pi\) (T) whenever \(\lambda\) is in the essential spectrum of T.