发表机构
Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明作用于可分复希尔伯特空间的I_k型冯·诺依曼代数中的每个算子均可分解，进而推出这类算子具有范数收敛性。
AI 中文摘要
本文研究作用于可分复希尔伯特空间的I_k型冯·诺依曼代数𝒯中算子的范数收敛性，即归一化幂序列在范数拓扑下的收敛性。通过酉共轭下的连续上三角形式，我们构造特定投影值族以证明I_k型冯·诺依曼代数中的每个算子都是可分解的，由此立即推出这类算子均具有范数收敛性。
英文摘要
Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$. For $A \in \mathcal{B}(\mathcal{H})$, we refer to the sequence $\{|A^{n}|^{1/n}\}_{n\in\mathbb{N}}$ as the normalized power sequence of $A$. In this article, we study the norm convergence property, i.e. convergence of normalized power sequence in the norm topology for operators belonging to type $\mathrm{I}_k$ von Neumann algebras acting on a separable complex Hilbert space. By utilizing continuous upper-triangular forms via unitary conjugations, we construct specific projection-valued families to prove that every operator in a type $\mathrm{I}_k$ von Neumann algebra is decomposable. As a consequence, this immediately establishes that every such operator possesses the norm convergence property, extending recent results known for matrices with complex-valued entries, compact operators on a separable Hilbert space, spectral operators, and Riesz operators. Finally, we provide a counterexample within the type $I_\infty$ factor $\mathcal{B}(l^2(\mathbb{N}))$ to demonstrate that this convergence property generally fails when the dimension $k$ is infinite.
