What does this research mean for the field?

The ideal structure, boundary quotient, K-theory, and index theory of the localized Bost-Connes system are fundamentally governed by the structural feature that the range projections of its prime isometries are nested rather than orthogonal. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Characterize the ideal structure, boundary quotient, K-theory, and index theory of localized Bost-Connes systems governed by free abelian semigroups on primes.

September 2, 2026Open Access

Localized Bost-Connes systems II: ideal structure, boundary quotient, and index

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Authors

RCRuqing ChenZhejiang Normal University

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Implication

Theoretical analysis reveals how nested range projections govern ideal structures and index theory in localized Bost-Connes systems, highlighting symmetry breaking.

Key Points

Characterize the ideal structure, boundary quotient, K-theory, and index theory of localized Bost-Connes systems governed by free abelian semigroups on primes.
Analyzed the topological dynamics of the underlying groupoid, evaluating isotropy on closed orbits and dual torus subsets.
Computed boundary quotients and K-groups via tight quotients and a collapsing Koszul spectral sequence for prime sets.
Investigated conditional expectations and subfactors at the symmetric KMS state, contrasting localized systems with affine systems featuring Cuntz relations.
The tight boundary quotient is a Type I matrix algebra over the group C*-algebra of the isotropy group, with the narrow class number as its sole isomorphism invariant.
For a single prime, K1 vanishes and K0 is free, whereas multiple primes yield K-theory as the abutment of a collapsing Koszul spectral sequence.
Inclusions of prime algebras exhibit infinite index at the symmetric KMS state without Galois correspondence, whereas finite index arises exclusively via symmetry breaking with relative entropy matching the logarithm of the index.

Cite This Study

Ruqing Chen (2026) studied this question.

synapsesocial.com/papers/6a97e2c5c562ede874ec7104 https://doi.org/10.5281/zenodo.22177725

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Implication

Theoretical analysis reveals how nested range projections govern ideal structures and index theory in localized Bost-Connes systems, highlighting symmetry breaking.

Localized Bost-Connes systems II: ideal structure, boundary quotient, and index

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Member takes

Implication

Key Points

Cite This Study

Localized Bost-Connes systems II: ideal structure, boundary quotient, and index

Discussion

Member takes

Implication

Key Points

Cite This Study

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