On the fixed point set of nonexpansive order preserving maps (Q1113442)

If T is a nonexpansive map of \(L^+_ 1\) of a probability space with \(Tc=c\) for every constant \(c\geq 0\), then T is order preserving, and for each f in \(L^+_ 1\) the sequence of Cesaro averages, \((1/N)(f+Tf+...+T^{N-1}f)\), converges weakly. We show that the limit need not be a fixed point for T, and that the set of fixed points need not be convex. Next we study a nonexpansive order preserving map T on \(L^+_{\infty}\) (or all of \(L_{\infty})\), and show that the set of fixed points, \(F=\{f:\) \(Tf=f\}\), is a lattice in the induced order. We show the existence of a nonexpansive order preserving retraction onto F which commutes with T.