Renormings and the fixed point property in non-commutative L1L1-spaces

Let MM be a finite von Neumann algebra. It is known that L1(M)L1(M) and every non-reflexive subspace of L1(M)L1(M) fail to have the fixed point property for non-expansive mappings (FPP). We prove a new fixed point theorem for this class of mappings in non-commutative L1(M)L1(M) Banach spaces which lets us obtain a sufficient condition such that a closed subspace of L1(M)L1(M) can be renormed to satisfy the FPP. As a consequence, we deduce that the predual of every atomic finite von Neumann algebra can be renormed with the FPP.