Quantum integrals II: Dominated and almost uniform convergence

Abstract

One main aim of our paper is to formulate noncommutative versions of the Lebesgue dominated convergence theorem in the setting of quantum integration theory on von Neumann algebras. Our starting point is a reformulation of classical dominated convergence in terms of weak* convergence. Using our recent sequential-continuity theorem, we obtain dominated convergence formulations for sequentially normal states and, under domination by a fixed bounded integrable majorant, for sequentially normal weights. Integrable, possibly unbounded, majorants are treated first in the tracial setting, then for arbitrary von Neumann algebras using Haagerup's L1 space, and finally using normal semifinite weights and their GNS representations. For any faithful normal semifinite weight $ϕ$, weak* convergence of norm-bounded sequences is characterized by `integrating against' bounded positive integrable majorants. We also study noncommutative almost uniform and local almost uniform convergence. In particular, for norm-bounded sequences, local $ϕ$-bilaterally almost uniform convergence implies weak* convergence and L1 convergence to 0 of $h^{1/2}(x_n-x)h^{1/2}$ for every positive Haagerup L1 density h, without assuming sigma-finiteness. We also prove a dominated convergence theorem in the predual: fixed positive order domination and Petz's almost uniform convergence imply convergence in the predual norm. We establish implications and equivalences among these convergence notions, connecting dominated convergence with tracial and nontracial integration.