Decomposability of Operators in Type Ik von Neumann Algebras
Abstract
Let ℋ be a complex Hilbert space and ℬ(ℋ) be the algebra of all bounded linear operators on ℋ. For A ∈ ℬ(ℋ), we refer to the sequence {|An|1/n}n∈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 Ik 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 Ik 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∞ factor ℬ(l2(ℕ)) to demonstrate that this convergence property generally fails when the dimension k is infinite.
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BibTeX
@misc{karri2026decomposability,
title = {{Decomposability of Operators in Type $\mathrmI_k$ von Neumann Algebras}},
author = {Ajay Kumar Karri},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18447},
eprint = {2608.18447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18447},
}
RIS
TY - UNPB
TI - Decomposability of Operators in Type I_k von Neumann Algebras
AU - Ajay Kumar Karri
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.18447
UR - https://arxiv.org/abs/2608.18447
AB - Let ℋ be a complex Hilbert space and ℬ(ℋ) be the algebra of all bounded linear operators on ℋ. For A ∈ ℬ(ℋ), we refer to the sequence {|A^n|^1/n}_n∈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 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 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_∞ factor ℬ(l^2(ℕ)) to demonstrate that this convergence property generally fails when the dimension k is infinite.
ER -