Property (SP) and Inclusions of C*-algebras
Abstract
As a generalization of Gabe and Neagu's inclusions of real rank zero, we introduce a notion of Property (SP) for inclusions of C*-algebras. We begin the systematic study of SP-inclusions, establishing a topological description in the commutative setting and permanence under many constructions. We also extend known results about the permanence of Property (SP) for C*-algebras under classical symmetries to the setting of quantum symmetries; we show that Property (SP) for (simple, separable) C*-algebras is preserved under irreducible C*-discrete inclusions. Finally, we resolve (in the negative) a question raised by Gabe and Neagu about inclusions coming from Furstenburg boundaries. Namely, we show the existence of a countable, discrete, non-amenable group G such that Cr*(G) ⊆ C(∂F G) rtimesr G is not an SP-inclusion (and therefore not an inclusion of real rank zero).
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BibTeX
@misc{gow2026property,
title = {{Property (SP) and Inclusions of C*-algebras}},
author = {Alec Gow and Roberto Hernández Palomares},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30057},
eprint = {2608.30057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30057},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Property (SP) and Inclusions of C*-algebras
AU - Alec Gow
AU - Roberto Hernández Palomares
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30057
UR - https://arxiv.org/abs/2608.30057
AB - As a generalization of Gabe and Neagu's inclusions of real rank zero, we introduce a notion of Property (SP) for inclusions of C*-algebras. We begin the systematic study of SP-inclusions, establishing a topological description in the commutative setting and permanence under many constructions. We also extend known results about the permanence of Property (SP) for C*-algebras under classical symmetries to the setting of quantum symmetries; we show that Property (SP) for (simple, separable) C*-algebras is preserved under irreducible C*-discrete inclusions. Finally, we resolve (in the negative) a question raised by Gabe and Neagu about inclusions coming from Furstenburg boundaries. Namely, we show the existence of a countable, discrete, non-amenable group G such that C_r^*(G) ⊆ C(∂_F G) rtimes_r G is not an SP-inclusion (and therefore not an inclusion of real rank zero).
ER -