PreprintCC BY-NC-ND 4.0arXiv
The Ulam--Kaplansky problem in operator algebras
Abstract
The Ulam--Kaplansky problem up to similarity in operator algebras asks whether A⊕ A∼ B⊕ B implies A∼ B. We construct block-diagonal compact operators for which this implication fails. For von Neumann algebras with separable predual, we prove that this cancellation property is equivalent to subhomogeneity. Moreover, we obtain a cancellation law for finite ampliations in unital Banach algebras when the commutant of one element is finite dimensional modulo its Jacobson radical.
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BibTeX
@misc{jiang2026ulamkaplansky,
title = {{The Ulam--Kaplansky problem in operator algebras}},
author = {Chunlan Jiang and Minghui Ma and Rui Shi and Yuanhang Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.20593},
eprint = {2607.20593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20593},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Ulam--Kaplansky problem in operator algebras
AU - Chunlan Jiang
AU - Minghui Ma
AU - Rui Shi
AU - Yuanhang Zhang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2607.20593
UR - https://arxiv.org/abs/2607.20593
AB - The Ulam--Kaplansky problem up to similarity in operator algebras asks whether A⊕ A∼ B⊕ B implies A∼ B. We construct block-diagonal compact operators for which this implication fails. For von Neumann algebras with separable predual, we prove that this cancellation property is equivalent to subhomogeneity. Moreover, we obtain a cancellation law for finite ampliations in unital Banach algebras when the commutant of one element is finite dimensional modulo its Jacobson radical.
ER -