Abstract
We provide characterizations of convex, compact for the topology of local convergence in measure subsets of non-commutative L1-spaces previously considered for classical L1-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∗→L1(M,τ) is the non-commutative Yosida–Hewitt projection, and C is a norm bounded subset of L1(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.
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