Revisiting the Transfinite Christensen-Pedersen Argument
Abstract
Christensen and Pedersen proved that every properly infinite AW*-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of AW*-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a *-homomorphism between AW*-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about κ-join-preserving *-homomorphisms. We also show that a finite AW*-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.
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BibTeX
@misc{arulseelan2026revisiting,
title = {{Revisiting the Transfinite Christensen-Pedersen Argument}},
author = {Jananan Arulseelan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20718},
eprint = {2609.20718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20718},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Revisiting the Transfinite Christensen-Pedersen Argument
AU - Jananan Arulseelan
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.20718
UR - https://arxiv.org/abs/2609.20718
AB - Christensen and Pedersen proved that every properly infinite AW^*-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of AW^*-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a *-homomorphism between AW^*-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about κ-join-preserving *-homomorphisms. We also show that a finite AW^*-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.
ER -