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

We provide characterizations of convex, compact for the topology of local convergence in measure subsets of non-commutative L 1 -spaces previously considered for classical L 1 -spaces. More precisely, if M is a semifinite and σ -finite von Neumann algebra equipped with a distinguished semifinite faithful normal trace τ , P : M ∗ → L 1 ( M , τ ) is the non-commutative Yosida–Hewitt projection, and C is a norm bounded subset of L 1 ( M , τ ) that is convex and closed for the topology of local convergence in measure then we isolate the precise conditions on C for which P : C ¯ w ∗ → C is compactness preserving, sequentially continuous, or continuous when C ¯ w ∗ is equipped with the weak* topology and C with the topology of local convergence in measure.