Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras

The classical Lebesgue dominated convergence theorem (DCT) is a cornerstone in commutative measure theory, while its extension to the noncommutative setting has remained a longstanding challenge due to the barriers posed by the noncommutative structure. Let \(α\) be a unitarily invariant norm defined on a diffuse tracial von Neumann algebra \((\mathcal{M}, τ)\), let $L^α(\mathcal M, τ)$ be the completion of $\mathcal M$ with respect to the norm $α$(which includes all $L^p(\mathcal M, τ)$ as special cases). In this paper, by replacing ``almost everywhere convergence" with Nelson's convergence in measure, we establish a noncommutative analog of the dominated convergence theorem for $L^α(\mathcal M, τ)$ under a { significantly weaker dominating condition} than those previously considered. For instance, in a \(II_1\) factor \(\mathcal{M}\), if \(P\) is a projection with \(τ(P)=\frac{1}{n}\), then \(n^2 P\) ``dominates" every operator of norm \(\leq n\). As an application, we obtain a complete characterization of center-preserving $*$-automorphism of finite von Neumann algebras, which unifies and extends the earlier works of Kadison and Haagerup on the structure of automorphisms.