Second dual projection characterizations of three classes of L0L0-closed, convex, bounded sets in L1L1: Non-commutative generalizations

We provide characterizations of convex, compact for the topology of local convergence in measure subsets of non-commutative L1L1-spaces previously considered for classical L1L1-spaces. More precisely, if MM is a semifinite and σσ-finite von Neumann algebra equipped with a distinguished semifinite faithful normal trace ττ, P:M∗→L1(M,τ)P:M∗→L1(M,τ) is the non-commutative Yosida–Hewitt projection, and CC is a norm bounded subset of L1(M,τ)L1(M,τ) that is convex and closed for the topology of local convergence in measure then we isolate the precise conditions on CC for which P:C¯w∗→C is compactness preserving, sequentially continuous, or continuous when C¯w∗ is equipped with the weak* topology and CC with the topology of local convergence in measure.