Maximal Rosenthal transfer principles and complete moment criteria for weighted triangular arrays of measurable operators

This paper studied complete moment convergence and polynomially weighted summability criteria for scalar weighted triangular arrays of measurable operators in tracial von Neumann algebras. The main result was a maximal Rosenthal transfer theorem for weighted triangular arrays whose rows admitted martingale filtrations. Starting from a uniform $ q $-moment estimate for each terminal row sum, it controlled all weighted partial sums in $ L_q(M; \ell_\infty) $ and, for $ q > 2 $, in $ L_q(M; \ell_\infty^{\mathrm c}) $. Under the associated scalar summability condition, it yielded row-uniform bilateral almost uniform convergence and, for $ q > 2 $, row-uniform almost uniform convergence. As an application of the general Rosenthal-type criterion, we derived a high-moment Marcinkiewicz–Zygmund normalization for freely independent rows and martingale-difference rows satisfying a conditional variance domination. The terminal estimates were verified directly from the inequalities of Junge and Xu. The Rosenthal summability criterion was then developed as a technical tool: Under explicit summability conditions on scalar weights and normalizing constants, together with uniform second and $ q $th moments such as those supplied by stochastic domination, it yielded complete $ p $-moment convergence for all $ 0 < p < q $. We also clarified the equivalent positive-part formulation of complete moment convergence and included a sharpness benchmark for freely independent semicircular rows, showing that the scalar variance summability condition was necessary and sufficient at the $ q $-moment summability level. The resulting weighted triangular-array criterion complemented sharp Baum–Katz theorems for successively independent or adapted sequences. Finally, we formulated a tracial C$ ^* $-probability version through the Gelfand–Naimark–Segal (GNS) representation and the generated von Neumann algebra.