Dominance and equivalence for states on C*-algebras: Quasi-Invariant states
Abstract
We study the noncommutative generalization of measure-theoretic dominance and equivalence of states on C*-algebras to explore quasi-invariance under group actions. For a dominated state, we derive an unbounded "Radon-Nikodym" derivative affiliated with the commutant algebra of the dominating state's GNS representation. Interestingly, this dominance is generally non-transitive because the product of the corresponding closed operators can be non-closable. When looking at group actions by *-automorphisms, the orbit of a fixed quasi-invariant state consists entirely of mutually equivalent states. However, the orbit closure may contain singular states, meaning the set of quasi-invariant states is closed under convex combinations but not topologically closed. The paper also provides a unitary implementation of the group action on the GNS Hilbert space-generalizing covariant representations and compares this approach with the Pedersen-Takesaki construction, where the Radon-Nikodym derivative sits in the centraliser instead of the commutant.
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BibTeX
@misc{dhahri2026dominance,
title = {{Dominance and equivalence for states on $C^*$-algebras: Quasi-Invariant states}},
author = {Ameur Dhahri and Francesco Fidaleo and Chul Ki Ko and Hyun Jae Yoo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00409},
eprint = {2609.00409},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00409},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dominance and equivalence for states on C^*-algebras: Quasi-Invariant states
AU - Ameur Dhahri
AU - Francesco Fidaleo
AU - Chul Ki Ko
AU - Hyun Jae Yoo
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00409
UR - https://arxiv.org/abs/2609.00409
AB - We study the noncommutative generalization of measure-theoretic dominance and equivalence of states on C^*-algebras to explore quasi-invariance under group actions. For a dominated state, we derive an unbounded "Radon-Nikodym" derivative affiliated with the commutant algebra of the dominating state's GNS representation. Interestingly, this dominance is generally non-transitive because the product of the corresponding closed operators can be non-closable. When looking at group actions by *-automorphisms, the orbit of a fixed quasi-invariant state consists entirely of mutually equivalent states. However, the orbit closure may contain singular states, meaning the set of quasi-invariant states is closed under convex combinations but not topologically closed. The paper also provides a unitary implementation of the group action on the GNS Hilbert space-generalizing covariant representations and compares this approach with the Pedersen-Takesaki construction, where the Radon-Nikodym derivative sits in the centraliser instead of the commutant.
ER -