Fixed point property and the Fourier algebra of a locally compact group

We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) spaces and use this for the Fourier algebra of a locally compact group In particular we show that if is an IN-group, then has the weak fpp if and only if is compact. We also show that if is any locally compact group, then has the fixed point property (fpp) if and only if is finite. Furthermore if a nonzero closed ideal of has the fpp, then must be discrete.