{"id":"1d70e6ef-6b9a-47ed-a73a-5d9d52af44b0","arxiv_id":"2509.19148","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Regular inclusions with a faithful invariant pseudo-expectation have their regular ideals determined by invariant regular ideals of the subalgebra, and quotients by regular ideals preserve the pseudo-Cartan property and its Cartan envelope.","lead":"This paper describes the regular ideals of a C*-algebra inclusion in terms of the smaller algebra, and shows that dividing by such an ideal preserves the pseudo-Cartan structure and its Cartan envelope. It gives operator algebraists a cleaner dictionary between an inclusion and its quotients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified — Theorem 3.11 is sound under its explicit ideal-intersection hypothesis, which is the genuinely load-bearing condition.","rationale":"The reader's weakest_assumption identifies exactly the condition that the proof of Theorem 3.11 needs: absence of nonzero regular ideals with trivial intersection with B. I reread the proof and found no hidden circularity or unjustified leap that would invalidate the central claim under the stated hypotheses. The structure-projection arguments in Proposition 3.10 are compressed but standard, and the citations to Exel [7] and the previous paper [5] supply the known Boolean-isomorphism backbone. Since the theorem is explicitly conditional on the ideal intersection property and the paper correctly notes in Remark 3.12 that the weaker regular ideal intersection property suffices, the ACCEPT verdict remains appropriate.","tokens_in":18523,"tokens_out":36371,"duration_ms":1032072,"concrete_test":"Construct an inclusion satisfying all hypotheses of Theorem 3.11 except the ideal intersection property, with a nonzero regular ideal J' satisfying J'∩B={0} (e.g., using the ideas of [22, Example 4.3] or [5, Section 7]); then check whether J'↦J'∩B is injective. This would settle whether the assumption is essential and whether the theorem's inverse formula can hold beyond its stated scope.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The one condition on which the central claim genuinely rests is the (regular) ideal intersection property. Both halves of the proof of Theorem 3.11 manufacture a nonzero regular ideal L with L∩B={0} (L=J∩L_K^⊥ or L=L_K∩J^⊥) and rule it out by this assumption. Without it, J↦J∩B can fail to be injective and the inverse formula L_K=K^{⊥⊥} loses its meaning. The paper states this hypothesis plainly, and Remark 3.12 correctly notes that only the regular-ideal version is used. For pseudo-Cartan inclusions the property follows from Definition 2.5 via the composite (A,D^c) and (D^c,D). I found no internal inconsistency or unproved nonstandard step that would threaten the theorem under the stated hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper continues the authors' study of regular ideals in inclusions of C*-algebras. For a regular inclusion B ⊆ A satisfying the ideal intersection property and admitting an N-invariant faithful pseudo-expectation Φ: A → I(B), Theorem 3.11 asserts that J ↦ J∩B is a Boolean isomorphism from Reg(A) onto the invariant regular ideals of B, with inverse K ↦ L_K = {a ∈ A : Φ(a*a) ∈ ι(K)^{⊥⊥}}. This generalizes [5, Theorem 3.24] from conditional expectations to pseudo-expectations. The authors then use this description to prove Theorem 3.16: for a pseudo-Cartan inclusion (A,D) with Cartan envelope (A1,D1,α), the lattices Reg(A), RegInv_A(D), RegInv_{A1}(D1), and Reg(A1) are all isomorphic. Section 4 gives necessary and sufficient conditions, Theorem 4.2, for a quotient by an ideal to preserve the faithful unique pseudo-expectation property, with corollaries for regular inclusions and abelian inclusions, and proves Theorem 4.6 that quotients of pseudo-Cartan inclusions by regular ideals are again pseudo-Cartan, with an explicit description of the Cartan envelope of the quotient.","tokens_in":18701,"tokens_out":25953,"duration_ms":229131,"significance":"These results provide an explicit, computable inverse for Exel's lattice isomorphism and extend the earlier work of the authors from conditional expectations to the broader pseudo-expectation framework. The pseudo-Cartan quotient theorem is a substantial generalization of [5, Theorem 4.8]. The paper is careful with its hypotheses: the ideal intersection property is stated plainly, and Remark 3.12 correctly notes that only the regular-ideal version is used. The arguments rely on established tools — Hamana's injective envelopes, Exel's theorem, and Pitts's pseudo-Cartan machinery — and are mostly detailed. The explicit inverse formula for L_K and the quotient preservation criteria are likely to be useful for future investigations of ideal structure in C*-algebras, especially in settings where conditional expectations are not available.","major_comments":[{"comment":"In the proof of the equality L_K = J_K^{⊥⊥}, after defining L_{K^⊥}, the text states: 'Since a ∈ L_{K^⊥}, we also have that ac ∈ L_{K^⊥}.' This step requires L_{K^⊥} to be at least a right ideal. By Proposition 3.10(ii), that holds if K^⊥ is N-invariant. The N-invariance of K^⊥ is not proved in the manuscript; it follows for regular inclusions from condition (inv), for instance via [7, Proposition 4.2], but the argument should be supplied. As written, the proof that L_K is a regular ideal is incomplete, and this is a load-bearing point for Theorem 3.11.","section":"Proposition 3.10(iii)"}],"minor_comments":[{"comment":"The displayed inclusion 'α(n)^* α(J)^⊥ α(n)^* ⊆ α(J)^⊥' appears to have a typo: the final 'α(n)^*' should be 'α(n)', i.e. α(n)^* α(J)^⊥ α(n) ⊆ α(J)^⊥.","section":"Theorem 3.15"},{"comment":"When concluding that J_K is a right ideal from the fact that an ∈ J_K for n in the generating semigroup N, the proof should explicitly note that J_K is closed under norm limits; this is immediate from continuity of Φ and closedness of ι(K), but it is used and should be stated.","section":"Proposition 3.10(i)"},{"comment":"The notation K1 is used in the proof before it is formally introduced in the theorem statement. It would improve readability to define K1 = α(J∩D)^{⊥⊥ D1} in the display following the statement, alongside the definition of J1.","section":"Theorem 4.6"}],"recommendation":"minor_revision","confidential_remarks":"I agree with the reader's overall positive assessment. The only point I would ask the authors to address is the missing justification of N-invariance for K^⊥ in Proposition 3.10(iii); this is a local, repairable gap and does not affect the truth of the main theorems. I do not see a circularity concern: the paper uses prior results as established tools, and the new theorems are genuine extensions. The manuscript is well within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a competent and useful next chapter in the authors' study of regular ideals, and it delivers on its main promises. The headline result, Theorem 3.11, replaces conditional expectations with faithful invariant pseudo-expectations: for a regular inclusion with the ideal intersection property, the map J ↦ J∩B is a Boolean isomorphism from Reg(A) onto the invariant regular ideals of B, and the inverse is explicitly given by K ↦ L_K = {a : Φ(a*a) ∈ ι(K)^{⊥⊥}}. That explicit inverse is a real improvement over Exel's existence theorem and over the conditional-expectation version in [5].\n\nThe paper does several things well. Theorem 3.16 shows the regular ideal lattice of a pseudo-Cartan inclusion is isomorphic to that of its Cartan envelope, and Theorem 4.6 proves the quotient of a pseudo-Cartan inclusion by a regular ideal is again pseudo-Cartan, with the Cartan envelope of the quotient being the quotient of the original Cartan envelope. That last result considerably generalizes [5, Theorem 4.8] and is the right payoff for the machinery. Theorem 4.2, giving necessary and sufficient conditions for the faithful unique pseudo-expectation property to survive quotients, is also clean and useful (Corollaries 4.3 and 4.4 are the immediate applications). The proofs are detailed and mostly transparent, and the use of prior results—Exel, Pitts, Hamana, their own [5]—is appropriate and not circular.\n\nThe soft spots are minor. The ideal intersection property is the genuinely load-bearing hypothesis, and the paper is explicit about this, even noting in Remark 3.12 that only the regular-ideal version is used. That is honest. The one place I wanted more detail is Proposition 3.10(i): the argument that a ∈ J_K implies an ∈ J_K for a normalizer n is compressed, especially the step using N-invariance of Φ and K to conclude Φ(n*a*an) ∈ ι(K). It works, but the exposition could be expanded there. Also, Theorem 4.2's condition J = L_K is crisp but L_K is only a left ideal in general; the paper notes this, so it is not a flaw, just a structural feature.\n\nI found no internal inconsistency or unproved step that threatens the central claims. The proof skeleton is coherent and the quotient theorems hold together. This paper is for operator algebraists working on Cartan and pseudo-Cartan inclusions and ideal structure; it deserves a serious referee, with the main effort going into checking Proposition 3.10 and Theorem 4.2 in detail. My verdict: send it out, and expect a minor revision for exposition rather than a mathematical overhaul.","headline":"A solid continuation of the authors' regular-ideal program: Theorem 3.11 gives a clean explicit inverse for the ideal-intersection isomorphism, and the quotient results are a genuine step beyond their earlier work.","tokens_in":19205,"tokens_out":1758,"would_cite":true,"duration_ms":16177,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A pseudo-expectation formula completely classifies the regular ideals of a regular C*-algebra inclusion.","keywords":["regular inclusions","C*-algebras","regular ideals","pseudo-expectations","ideal intersection property","pseudo-Cartan inclusions","Cartan envelope","quotients"],"falsifier":"Test the inverse formula in a concrete regular inclusion, for example a higher-rank graph algebra C*(Λ) with its cycline subalgebra M_Λ from Example 2.12: take a nontrivial invariant regular ideal K of M_Λ, compute L_K = {a : Φ(a*a) ∈ ι(K)^{⊥⊥}}, and check whether L_K is a two-sided ideal of C*(Λ) with L_K ∩ M_Λ = K. Any K for which L_K fails to be an ideal, or for which the intersection property fails, would contradict Theorem 3.11.","tokens_in":18417,"feed_emoji":"🧮","tokens_out":7740,"duration_ms":63129,"temperature":0.7,"pith_summary":"This paper proves that for a regular inclusion of C*-algebras satisfying the ideal intersection property and carrying a faithful invariant pseudo-expectation, every regular ideal of the larger algebra is determined by its intersection with the smaller algebra together with the pseudo-expectation. The inverse map is explicit: an invariant regular ideal K of the subalgebra is sent to {a : Φ(a*a) ∈ ι(K)^{⊥⊥}}. This yields a Boolean algebra isomorphism between the regular ideals of A and the invariant regular ideals of B. The authors then show that quotients by regular ideals preserve the faithful unique pseudo-expectation property, and for pseudo-Cartan inclusions the quotient is again pseudo-Cartan with its Cartan envelope being the quotient of the original Cartan envelope. These results unify and extend earlier work on Cartan inclusions and graph algebras.","feed_headline":"Pseudo-expectation formula completely classifies regular ideals","feed_subtitle":"The ideal structure of a regular inclusion is captured by invariant ideals of the subalgebra plus one map.","key_machinery":"The engine is the pseudo-expectation Φ : A → I(B), a completely positive extension of the inclusion into the injective envelope of B, together with its invariance under the partial dynamics induced by normalizers. Regular ideals are characterized by J = J^{⊥⊥}; the inverse formula L_K uses the annihilator K^{⊥⊥} inside B and pulls it back through Φ via Φ(a*a). Structure projections in I(B) connect regular ideals of B with hereditary subalgebras, enabling the quotient arguments.","core_discovery":"The central result is Theorem 3.11: for a regular inclusion (A,B) with the ideal intersection property and a faithful pseudo-expectation Φ that is invariant under a generating semigroup of normalizers of B, the map J ↦ J∩B is a Boolean algebra isomorphism from the regular ideals of A onto the invariant regular ideals of B. The inverse is given explicitly: an invariant regular ideal K of B is sent to L_K = {a ∈ A : Φ(a*a) ∈ ι(K)^{⊥⊥}}, which also equals K^{⊥⊥}. For a pseudo-Cartan inclusion (A,D), this is combined with a comparison of invariant ideals under the Cartan envelope to show that the regular ideal lattices of A, D, the Cartan envelope A₁, and its diagonal D₁ are all isomorphic. The","pith_inferences":["The explicit inverse formula suggests a concrete computational route to regular ideals in examples such as higher-rank graph algebras: compute the invariant regular ideals of the diagonal or cycline algebra and apply Φ; Example 2.12 is a natural testbed.","The Boolean algebra isomorphism between Reg(A) and RegInv_A(B) may lift to an isomorphism of primitive ideal spaces or K-theoretic invariants, a direction the paper does not explore.","Theorem 4.2 isolates exactly when the faithful unique pseudo-expectation property survives quotienting; this could be used to characterize regular ideals as the 'quotient-compatible' ideals for regular inclusions, potentially connecting to boundary ideals and reduced crossed products.","For inclusions failing the ideal intersection property, the failure of the isomorphism is measured by 'floating' regular ideals having zero intersection with B; studying these could yield a quantitative invariant of how far an inclusion is from being pseudo-Cartan."],"forward_implications":["If the paper is correct, the regular ideal lattice of a regular inclusion is completely determined by the invariant regular ideals of the subalgebra and the pseudo-expectation; no additional data are needed.","For pseudo-Cartan inclusions, the regular ideal lattice is invariant under taking the Cartan envelope, so ideal-theoretic questions can be transferred to the better-understood Cartan setting.","Quotienting a regular inclusion by a regular ideal preserves the faithful unique pseudo-expectation property (Corollary 4.3), and the same holds for inclusions of abelian C*-algebras (Corollary 4.4).","The quotient of a pseudo-Cartan inclusion by a regular ideal is again pseudo-Cartan, and the Cartan envelope of the quotient is exactly the quotient of the original Cartan envelope (Theorem 4.6).","Theorem 4.2 gives a precise criterion for when an arbitrary ideal quotient preserves faithfulness of the unique pseudo-expectation: the ideal must equal {a : Φ(a*a) ∈ ι(J∩B)^{⊥⊥}}."],"fun_headline_variants":["Regular ideals: one map captures them all","Pseudo-expectation decodes regular ideal lattices","Invariant ideals + pseudo-expectation = full ideal structure","Quotients preserve Cartan: pseudo-expectation classifies ideals","Ideal classification via pseudo-expectation and Cartan quotients"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole description rests on the assumption that every nonzero regular ideal of A has nonzero intersection with B (the ideal intersection property, or its regular-ideal version); if a 'floating' regular ideal exists with no trace in B, the isomorphism and the inverse formula collapse.","fun_headline_variants_meta":{"raw":{"variants":["Regular ideals: one map captures them all","Pseudo-expectation decodes regular ideal lattices","Invariant ideals + pseudo-expectation = full ideal structure","Quotients preserve Cartan: pseudo-expectation classifies ideals","Ideal classification via pseudo-expectation and Cartan quotients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000161,"raw_usage":{"total_tokens":1041,"prompt_tokens":678,"completion_tokens":363,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":293}},"tokens_in":422,"tokens_out":363,"duration_ms":3673,"temperature":1.0,"reasoning_tokens":293,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:25:00.866296+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the inverse formula in a concrete regular inclusion, for example a higher-rank graph algebra C*(Λ) with its cycline subalgebra M_Λ from Example 2.12: take a nontrivial invariant regular ideal K of M_Λ, compute L_K = {a : Φ(a*a) ∈ ι(K)^{⊥⊥}}, and check whether L_K is a two-sided ideal of C*(Λ) with L_K ∩ M_Λ = K. Any K for which L_K fails to be an ideal, or for which the intersection property fails, would contradict Theorem 3.11.","supporting_citations":[],"review_version":1}