Fixed point properties of semigroups of nonexpansive mappings

We prove that if H is a Hilbert space then the Schatten (trace) class operators on H has the weak∗ fixed point property for left reversible semigroups. This answered positively a problem raised by A.T.-M. Lau. We also prove that if M is a finite von Neumann algebra then any nonempty bounded convex subset of the non-commutative L1-space associated to M that is compact for the measure topology has the fixed point property for left reversible semigroups.