PreprintCC BY 4.0arXiv
Comparisons of benchmark equivalence relations
Abstract
We determine the Borel-reducibility comparison of each pair of Π03 benchmark equivalence relations in Gao's monograph. The additional comparisons we prove are E∞≤B c0, E∞nleqBℓp for 2<p<∞, ℓpnleqB≡m for 2<p<∞, c0nleqB≡m, c0nleqBℓ∞+ and ≡mnleqBℓ∞+. Together with the previous positive direction, these give ℓp≤B≡m if and only if p≤2. These answer several questions in Gao's monograph and Kanovei' monograph. We also prove that unitary equivalence of all bounded linear operators is not Borel reducible to ≡m. This answers a question of Ding and Gao. Equivalently, unitary equivalence of arbitrary representations of F∞ is strictly more complicated than that of arbitrary representations of ℤ. This answers a question of Thomas.
Related
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BibTeX
@misc{wang2026comparisons,
title = {{Comparisons of benchmark equivalence relations}},
author = {Ruijun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35025},
eprint = {2610.35025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Comparisons of benchmark equivalence relations
AU - Ruijun Wang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35025
UR - https://arxiv.org/abs/2610.35025
AB - We determine the Borel-reducibility comparison of each pair of Π^0_3 benchmark equivalence relations in Gao's monograph. The additional comparisons we prove are E_∞≤_B c_0, E_∞nleq_Bℓ_p for 2<p<∞, ℓ_pnleq_B≡_m for 2<p<∞, c_0nleq_B≡_m, c_0nleq_Bℓ_∞^+ and ≡_mnleq_Bℓ_∞^+. Together with the previous positive direction, these give ℓ_p≤_B≡_m if and only if p≤2. These answer several questions in Gao's monograph and Kanovei' monograph. We also prove that unitary equivalence of all bounded linear operators is not Borel reducible to ≡_m. This answers a question of Ding and Gao. Equivalently, unitary equivalence of arbitrary representations of F_∞ is strictly more complicated than that of arbitrary representations of ℤ. This answers a question of Thomas.
ER -