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.
Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. A subalgebra $\mathfrak{A}$ in $\mathcal{B}(\mathcal{H})$ has the left (resp.\ right) partial factorization property if for any invertible operator $S\in\mathc...
Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathca...
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\mathcal{C}_N\mathfrak{g}=\mathfrak{g}\otimes (\mathbb{C}[t]/t^{N+1}\mathbb{C}[t])$ denote the $N$-th Takiff algebra of $\mathfrak{g}$. We introduce a notion of parabolic subalgebras of $\mathcal{C}_N\mathfrak{g}$ and classify them in t...
Let $W_{\geq -1} = \mathbb{C}[t]\partial$ and $W = \mathbb{C}[t, t^{-1}]\partial$ be the Witt algebra of algebraic vector fields on $\mathbb{C}$ and $\mathbb{C}^*$ respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(...
Let $LS(\mathcal{M})$ be the algebra of locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$, equipped with an $F$-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between $LS(\mathcal{M})$ admits a canonical representation of the...
Jing-Hao Huang, K. Kudaybergenov, Bing Yan· 0 citations
For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$ \operatorname{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_...
Rui-Ke Liu, Jie Shen· 0 citations
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