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A characterization of purely atomic von Neumann algebras
Abstract
Let M be a semifinite von Neumann algebra with a faithful normal semifinite trace τ, and let Φ be an Orlicz function. We prove that the noncommutative Orlicz space LΦ(M,τ) is contained in M if and only if M is purely atomic and there is δ>0 such that τ(e)≥δ for every nonzero projection e∈ M. We also prove that every left M-module map on LΦ(M,τ) is bounded. As a consequence, a left ideal of M contained in LΦ(M,τ) and dense in it admits an M-invariant algebraic complement if and only if it is equal to LΦ(M,τ). These results extend a theorem of Fidaleo and Zsidó from L2 to all noncommutative Orlicz spaces.
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BibTeX
@misc{de2026characterization,
title = {{A characterization of purely atomic von Neumann algebras}},
author = {Debabrata De},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07978},
eprint = {2611.07978},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07978},
}
RIS
TY - UNPB
TI - A characterization of purely atomic von Neumann algebras
AU - Debabrata De
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07978
UR - https://arxiv.org/abs/2611.07978
AB - Let M be a semifinite von Neumann algebra with a faithful normal semifinite trace τ, and let Φ be an Orlicz function. We prove that the noncommutative Orlicz space L^Φ(M,τ) is contained in M if and only if M is purely atomic and there is δ>0 such that τ(e)≥δ for every nonzero projection e∈ M. We also prove that every left M-module map on L^Φ(M,τ) is bounded. As a consequence, a left ideal of M contained in L^Φ(M,τ) and dense in it admits an M-invariant algebraic complement if and only if it is equal to L^Φ(M,τ). These results extend a theorem of Fidaleo and Zsidó from L^2 to all noncommutative Orlicz spaces.
ER -