{"id":"acc2cc4b-d4d0-4c52-ae3b-04f97f9420e2","arxiv_id":"2601.15204","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For p≠2, embeddings of reduced L^p-groupoid algebras are equivalent to groupoid morphisms, ruling out AF-embeddability of irrational rotation L^p-algebras and any L^p analog of Kirchberg's O_2 embedding theorem.","lead":"For p not equal to 2, the paper shows that embeddings between L^p-operator algebras attached to étale groupoids are rigid: they are forced to come from maps of the underlying groupoids. This yields concrete no-go results, including that irrational-rotation L^p-algebras cannot sit inside spatial AF L^p-algebras and that O_2^p ⊗_p O_2^p cannot sit inside O_2^p.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.20 collapses if the weak-* limit in Proposition 3.33 is not a true MP-partial isometry; the proof of Theorem 3.34 leaves this unverified.","rationale":"The reader and I agree on the load-bearing step. I am not claiming the theorem is false; the gap may be fillable. But as written, the proof of the key preservation theorem depends on a weak-* limit having MP-partial-isometry structure, and that structure is not closed under weak-* limits in general. Since Theorem 4.12 and hence Theorem 4.20 use φ(SN) to define the actor, this is the narrowest point at which the central claim can fail. I do not move the verdict because the condition already captures this risk; I would make the acceptance condition explicit: either supply the missing proof or cite a reference that contains it. Independent verification of [22, Cor 11.19] remains necessary for Corollary 7.9 but not for Theorem 4.20.","tokens_in":43371,"tokens_out":31621,"duration_ms":321960,"concrete_test":"Provide a complete proof of the missing algebraic identities for Φ: for u,v∈PI_X(A**) that appear in product representations, show that Φ(uv)=Φ(u)Φ(v) and that Φ(u†)=Φ(u)†, using the approximating nets and the realization identities of Definition 3.19; then verify the hermitian conditions by identifying the weak-* limits of φ(uv e_i) and φ(vu e_i) explicitly. Apply this to the simplest nontrivial case A=F_p^λ(G,Σ) with a non-trivial open bisection S, p≠2: compute Φ(1_S) from the definition and check directly that Φ(1_S)Φ(1_S^*) and Φ(1_S^*)Φ(1_S) are hermitian in B. If the computation is impossible without an extra hypothesis (e.g. φ is isometric), Theorem 3.34 must be weakened and Theorem 4.20 re-examined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 4.20, and its proof funnels through Theorem 3.34 (automatic preservation of spatial normalizers) and Theorem 3.35. The decisive step is Proposition 3.33: for a spatial normalizer n=uh with u∈PI_X(A**), Φ(u):=weak-*-lim φ(ue_i) is asserted to be an MP-partial isometry in B. The proof concludes: 'Φ inherits unitality and contractivity from φ and therefore preserves MP-partial isometries by Remark 3.13.' This is not an argument: Φ is a partially defined weak-* limit, not a homomorphism on A, and MP-partial-isometry identities (uvu=u, vuv=v, uv,vu∈B_h) are not automatic under weak-* limits. The paper never proves Φ(uv)=Φ(u)Φ(v) or Φ(u†)=Φ(u)†. If this fails, φ(uh) need not belong to SN_Y(B), the actor in Theorem 4.12 is not produced, and the equivalence in Theorem 4.20 has no foundation. The rest of the paper—including Corollary 5.6 and the conditional Corollary 7.9—depends on this step for p≠2. The unpublished dependence on [22, Cor 11.19] is a separate bottleneck for Corollary 7.9 only.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a rigidity theory for embeddings of L^p-operator algebras associated with étale groupoids. Its main result, Theorem 4.20, asserts that for p∈(1,∞)\\{2\\} and compact-unit Weyl twists, a unital contractive homomorphism between reduced twisted groupoid algebras that intertwines the canonical conditional expectations is necessarily induced by a groupoid-level actor diagram. The proof passes through a theory of spatial normalizers and core inclusions modeled on Renault's C*-algebraic reconstruction. The advertised applications are: (i) for p≠2, a reduced L^p-groupoid algebra of a principal Weyl groupoid embeds into a spatial AF L^p-algebra only if the groupoid is AF, giving non-embeddability of irrational rotation L^p-algebras; (ii) for p≠2, there is no unital contractive homomorphism from O_2^p ⊗_p O_2^p to O_2^p, so no L^p-analogue of Kirchberg's O_2-embedding theorem. The paper also shows that embeddings of L^p-groupoid algebras induce embeddings of topological full groups and identifies the relevant full groups with generalized Brin–Thompson groups.","tokens_in":43705,"tokens_out":15206,"duration_ms":151985,"significance":"If the main theorem is correct, the paper gives a striking and unexpected contrast with C*-algebra theory: in the L^p setting with p≠2, embeddings of reduced groupoid algebras are rigid enough to be described entirely by morphisms of the underlying groupoids. The applications to AF-embeddability and to Cuntz-algebra tensor products are substantial and falsifiable. The paper is well structured, carefully written, and contains a considerable amount of original technical machinery, including the spatial-normalizer action and the actor reinterpretation. However, the central technical step in Section 3 is not proved as written, and one auxiliary claim in Section 5 is false. These issues are load-bearing for the main results, so significant revision is required before the paper can be accepted.","major_comments":[{"comment":"In the proof of Proposition 4.19 the symbol 'supp′' appears without definition; it should be 'supp' or should be explicitly defined.","section":"§4, Proposition 4.19"}],"minor_comments":[{"comment":"The first line contains a typo: 'φ(core(A))⊆φ(core(B))' should read 'φ(core(A))⊆core(B)'.","section":"§3, Theorem 3.34"},{"comment":"The notation 'e∈φ(C0(U))B∗' for the weak-* closure is not defined; use an overline or a verbal description.","section":"§3, Proposition 3.33"},{"comment":"There is a typo in the introduction: 'Pismner-Voiculescu' should be 'Pimsner-Voiculescu'.","section":"§1, Introduction"},{"comment":"Reference [33] has a stray '2.' at the end of the arXiv identifier; reference [22] should be marked as an unpublished preprint with its current status.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially important, but the present version has a serious gap in the central technical proposition of Section 3 and a false auxiliary claim in Section 5. I would not recommend rejection if the authors can supply the missing argument for Proposition 3.33 and correct the AF proof, but the burden of proof is on them. The dependence on [22, Cor. 11.19] should also be clarified before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I’ll cut to the chase: this is a strong, ambitious paper, and the main theorem is a real advance—but there’s a proof gap in a load-bearing spot that needs fixing before I’d trust the applications.\n\nThe genuinely new content is Theorem 4.20: for p≠2, unital contractive homomorphisms between reduced L^p-groupoid algebras that preserve the canonical conditional expectation are exactly those induced by a free actor between Weyl twists. That extends the isometric-isomorphism rigidity of Hetland–Ortega to arbitrary embeddings, and it uses spatial normalizers and actor theory in a way that is genuinely different from the C*-setting. The corollaries are sharp: irrational rotation L^p-algebras don’t embed into spatial AF L^p-algebras, and there’s no unital contractive map from O_2^p ⊗_p O_2^p to O_2^p. The topological-full-group application to Brin–Thompson groups is a nice bonus, and the paper is well-written and well-organized.\n\nThe soft spot is Proposition 3.33. The proof defines Φ(u) as a weak-* limit of φ(ue_i) and then says that because Φ inherits unitality and contractivity, it preserves MP-partial isometries by Remark 3.13. That’s not an argument. The MP-conditions Φ(u)Φ(u†)Φ(u)=Φ(u), Φ(u†)Φ(u)Φ(u†)=Φ(u†), and the hermitianity of the products are not automatic under weak-* limits, and the proof never establishes that Φ is multiplicative on the relevant partial isometries. The earlier claims about nets in the proof show the limit is independent of the approximating net, but they don’t give multiplicativity. This matters because Theorem 3.34 uses Banach–Lamperti on Φ(u) to conclude it is a spatial partial isometry; without the MP-partial-isometry conclusion, that step fails, and Theorem 4.12 and the whole characterization collapse. It may well be fixable—one can imagine working with the algebra elements φ(uh) and canceling φ(h) on its support—but it’s not in the paper.\n\nA separate, lesser concern: Corollary 7.9 leans on an unpublished result [22, Cor. 11.19] for Brin–Thompson group rigidity. That’s a dependency, not a flaw, but it makes the O_2^p statement conditional.\n\nSpecialists in L^p-operator algebras and groupoid rigidity should read this paper, and it deserves a serious referee. I’d send it back asking for a repaired Proposition 3.33, and I wouldn’t cite the main theorem in my own work until that’s settled.","headline":"Strong and ambitious paper, but the proof of the key automatic-preservation step (Prop 3.33) has a real gap that the main theorem leans on.","tokens_in":44193,"tokens_out":6083,"would_cite":false,"duration_ms":60796,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A22","46H05","46L05","47L10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For p≠2, a unital contractive homomorphism between reduced L^p-algebras of Weyl twists is isometric and respects the diagonal exactly when it is induced by a groupoid morphism, and this rigidity rules out several embeddings that exist in th","keywords":["L^p-operator algebras","groupoid C*-algebras","Weyl twists","spatial normalizers","actors","AF-embeddability","Cuntz algebras","topological full groups"],"falsifier":"Find a p∈(1,∞), p≠2 and two Weyl twists (G,Σ),(H,Ω) with compact unit spaces for which there exists a unital isometric homomorphism φ:F_p^λ(G,Σ)→F_p^λ(H,Ω) with φ∘E_G=E_H∘φ that is not of the form ι_*∘π_* for any intermediate Weyl twist; Theorem 4.20 says no such map exists. A more local falsifier is a single unital contractive homomorphism in this setting that does not send one spatial normalizer to a spatial normalizer, which would refute the automatic-preservation theorem and the actor construction.","tokens_in":43278,"feed_emoji":"🧩","tokens_out":14529,"duration_ms":136577,"temperature":0.7,"pith_summary":"The paper tries to establish a rigidity theorem for L^p-operator algebras at p≠2: algebras built from twisted etale groupoids admit almost no 'soft' embeddings, because every reasonable embedding is forced to respect the groupoid structure underneath. The central claim is that a unital contractive homomorphism between reduced L^p-algebras of Weyl twists with compact unit spaces is isometric and respects the canonical projection onto the diagonal subalgebra if and only if it is induced by a diagram of groupoid morphisms—an 'actor'—so the embedding is really the shadow of a groupoid morphism. A sympathetic reader should care because this converts otherwise intractable embedding questions into concrete groupoid questions, and it leads to results that contradict the C*-world: irrational-rotation L^p-algebras do not embed into spatial AF L^p-algebras, and O_2^p ⊗_p O_2^p does not embed into O_2^p, so the classical O_2-embedding theorem of C*-algebra theory has no L^p analogue for p≠2.","feed_headline":"For p≠2, embeddings of L^p-groupoid algebras are groupoid maps","feed_subtitle":"Rigidity blocks irrational-torus embeddings into AF L^p-algebras and any L^p analogue of the O_2-embedding theorem.","key_machinery":"The two central objects are spatial normalizers and actors. A spatial normalizer is an element of the algebra that realizes a partial homeomorphism of the diagonal; it is built from an MP-partial isometry (the Banach-algebraic analogue of a partial isometry) in the double dual, and off p=2 the structure theorem for partial isometries on L^p-spaces identifies these with the spatial partial isometries. An actor between twists is a consistent lifting operation: a free action of one twisted groupoid on another that commutes with right multiplication, equivalently an inverse semigroup homomorphism of bisections plus an equivariant anchor map of unit spaces. The machinery does two things: Theorem","core_discovery":"The central discovery is Theorem 4.20. Let (G,Σ) and (H,Ω) be Weyl twists—twisted etale groupoids whose base is Hausdorff and effective—with compact unit spaces, and let p∈(1,∞), p≠2. A unital contractive homomorphism φ:F_p^λ(G,Σ)→F_p^λ(H,Ω) is isometric and intertwines the conditional expectations (equivalently, is injective on the diagonal C(X)) if and only if there is an intermediate Weyl twist (G·_h Y, Σ·_β Y) and twist homomorphisms π:Σ→Σ·_β Y (surjective, fiberwise bijective) and ι:Σ·_β Y→Ω (injective, unitwise bijective, open image) such that φ=ι_*∘π_*, i.e. φ(f) is the zero-extension of f∘π_β. In words: every embedding is a groupoid morphism in disguise. The proof first shows (Theore","pith_inferences":["Editorial inference: the sharp break at p=2 suggests an actual phase transition in embeddability rather than a technical gap; one could test whether the class of embeddable L^p-groupoid algebras shrinks continuously as p moves away from 2, or whether the transition is abrupt.","Editorial inference: the m≤n theorem does not say whether m≤n is sufficient; a natural test is to try to construct an actor from a product of m shifts of finite type to a product of n shifts whenever m≤n, which would show the bound is sharp.","Editorial inference: because the proof uses compact unit spaces, Hausdorffness, and effectiveness, extending the normalizer-preservation argument to non-Hausdorff or non-effective twists would expose which hypothesis is truly load-bearing; the main theorem predicts rigidity should fail or require new tools there."],"forward_implications":["Embeddability between reduced L^p-groupoid algebras for p≠2 becomes a groupoid problem: one looks for an actor diagram, and non-existence can often be certified by groupoid invariants alone.","A principal Weyl groupoid algebra embeds into a spatial AF L^p-algebra if and only if the underlying groupoid is AF; consequently the L^p irrational rotation algebra A_θ^p has no unital contractive embedding into a spatial AF L^p-algebra (Corollary 5.6).","An isometric embedding of L^p-groupoid algebras induces an embedding of the associated topological full groups (Theorem 6.11), opening a route from Banach-algebra rigidity to group rigidity.","For p≠2 there is no unital contractive map O_2^p ⊗_p O_2^p → O_2^p, so no L^p analogue of the classical O_2-embedding theorem exists; more generally, a unital contractive map between tensor products of L^p-Cuntz algebras forces m≤n (Theorem 7.8 and Corollary 7.9)."],"fun_headline_variants":["L^p embeddings are groupoid maps for p≠2","No L^p analogue of O_2 embedding theorem","Irrational L^p tori don't embed in AF algebras","Rigidity: L^p-groupoid embeddings are groupoid morphisms"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is the automatic-preservation theorem for p≠2: a unital contractive homomorphism must send the diagonal subalgebra and its normalizing partial isometries into the corresponding objects of the target; if that preservation fails for some pair of algebras, the induced actor—and with it the entire groupoid-level description—collapses.","fun_headline_variants_meta":{"raw":{"variants":["L^p embeddings are groupoid maps for p≠2","No L^p analogue of O_2 embedding theorem","Irrational L^p tori don't embed in AF algebras","Rigidity: L^p-groupoid embeddings are groupoid morphisms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000836,"raw_usage":{"total_tokens":3597,"prompt_tokens":971,"completion_tokens":2626,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":715,"completion_tokens_details":{"reasoning_tokens":2551}},"tokens_in":715,"tokens_out":2626,"duration_ms":18734,"temperature":1.0,"reasoning_tokens":2551,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T08:55:23.285757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a p∈(1,∞), p≠2 and two Weyl twists (G,Σ),(H,Ω) with compact unit spaces for which there exists a unital isometric homomorphism φ:F_p^λ(G,Σ)→F_p^λ(H,Ω) with φ∘E_G=E_H∘φ that is not of the form ι_*∘π_* for any intermediate Weyl twist; Theorem 4.20 says no such map exists. A more local falsifier is a single unital contractive homomorphism in this setting that does not send one spatial normalizer to a spatial normalizer, which would refute the automatic-preservation theorem and the actor construction.","supporting_citations":[],"review_version":1}