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) $\mathcal{L}^1$ spaces and use this for the Fourier algebra $A(G)$ of a locally compact group $G.$ In particular we show that if $G$ is an IN-group, then $A(G)$ has the weak fpp if and only if $G$ is compact. We also show that if $G$ is any locally compact group, then $A(G)$ has the fixed point property (fpp) if and only if $G$ is finite. Furthermore if a nonzero closed ideal of $A(G)$ has the fpp, then $G$ must be discrete.