Truncation bands, unitality, and homomorphisms in truncated Riesz spaces: Boulabiar's problems and beyond
Abstract
We answer three questions from Boulabiar's survey on truncated Riesz spaces. First, the truncation band projection property coincides with the principal projection property, with no completeness or Archimedean hypothesis needed, and the band projection onto a truncation band is then given explicitly by the truncation itself. Second, a truncation on any Riesz space is unital exactly when its fixed-point set has a supremum. Hence bounded truncations on KB-spaces are unital, and a nonunital truncated Banach lattice with bounded truncation contains a closed sublattice copy of c0 and is never an AL-space, while on the AM side the smallest truncation unitization norm is always an M-norm, though the largest one need not be. Third, a truncation homomorphism whose domain truncation is Archimedean need not be a Riesz homomorphism, as an explicit counterexample built from c0 and c0/c00 shows, but it is one modulo truncation infinitesimals whenever the codomain space is Archimedean.
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BibTeX
@misc{habibi2026truncation,
title = {{Truncation bands, unitality, and homomorphisms in truncated Riesz spaces: Boulabiar's problems and beyond}},
author = {Mohamed Habibi and Hamza Hafsi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26933},
eprint = {2609.26933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26933},
}
RIS
TY - UNPB
TI - Truncation bands, unitality, and homomorphisms in truncated Riesz spaces: Boulabiar's problems and beyond
AU - Mohamed Habibi
AU - Hamza Hafsi
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26933
UR - https://arxiv.org/abs/2609.26933
AB - We answer three questions from Boulabiar's survey on truncated Riesz spaces. First, the truncation band projection property coincides with the principal projection property, with no completeness or Archimedean hypothesis needed, and the band projection onto a truncation band is then given explicitly by the truncation itself. Second, a truncation on any Riesz space is unital exactly when its fixed-point set has a supremum. Hence bounded truncations on KB-spaces are unital, and a nonunital truncated Banach lattice with bounded truncation contains a closed sublattice copy of c_0 and is never an AL-space, while on the AM side the smallest truncation unitization norm is always an M-norm, though the largest one need not be. Third, a truncation homomorphism whose domain truncation is Archimedean need not be a Riesz homomorphism, as an explicit counterexample built from c_0 and c_0/c_00 shows, but it is one modulo truncation infinitesimals whenever the codomain space is Archimedean.
ER -