Kazhdan's Property (T) for the unitary group of aseparable Hilbert space

We show that the unitary group of a separable Hilbert space hasKazhdan's Property (T), when it is equipped with the strong operatortopology. More precisely, for every integer m2, we give anexplicit Kazhdan set consisting of m unitary operators and determinean optimal Kazhdan constant for this set. Moreover, we show that alocally compact group with Kazhdan's Property (T) has a finite Kazhdanset if and only if its Bohr compactification has a finite Kazhdanset. As a consequence, if a locally compact group with Property (T)is minimally almost periodic, then it has a finite Kazhdan set.