{"id":"099c545a-2d3d-4af3-9c9d-c3d3a72faf14","arxiv_id":"2412.11191","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In the Cohen model of set theory, the C*-algebra (c0(2^ω))^N does not embed into the Calkin algebra Q(ℓ2), although c0(2^ω) itself always does.","lead":"This paper proves that in the Cohen model of set theory the C*-algebra (c0(2^ω))^N, the countable product of copies of c0(2^ω), cannot be embedded into the Calkin algebra, the quotient of bounded operators on Hilbert space by compact operators. The result adds a new consistency example to the study of which C*-algebras embed into the Calkin algebra, using forcing methods that do not rely on rigidity of quotient structures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof is internally coherent and the forcing argument is standard. A machine check of Proposition 26 remains the only substantive verification step.","rationale":"The paper's strongest claim, Theorem 27, is supported by Proposition 26 and the surrounding lemmas. The reader identified Lemma 25 as the weakest assumption; I partially agree, but I do not see a concrete failure. Lemma 25 is a defensive use of CH that makes the non-symmetric names into symmetric ones by mapping the countable supports into N and forming an ω2-sized subfamily; this is a standard argument. The proof of Lemma 15 builds γξ with disjoint supports and is constructive. Lemma 19's symmetry condition for extending conditions is carefully checked. The inner-product inequality at the end if the forcing part can go wrong, but I could not find an actual algebra error: it combines (11), (12), (14), Lemma 11, and Cauchy-Schwarz. The one honest concern is that this is an intricate forcing proof with no machine-checked formalization, so the risk of a subtle error is real. However, 'not machine-checked' is a verification risk, not a discovered flaw. Rule 6 says that one may agree with a paper's internally-consistent argument even when pointing to a correctness risk; Rule 2 explicitly permits a non-finding. I therefore set verdict_should_be to UNCHANGED and recommend a formal verification as the concrete test. My agreement_with_reader is 'partial': the reader's weakest assumption is plausible as a risk locus, but I would not elevate it to a load-bearing concern because the relevant hypotheses (countable supports, Δ-system, CH) are explicitly built into the proof.","tokens_in":16649,"tokens_out":1517,"duration_ms":13170,"concrete_test":"Re-run the combinatorial core of Proposition 26 in a proof assistant: formalize Lemma 25, Lemma 15, Lemma 19, and the final inner-product inequality (the chain following (30)) in Lean or a similar system. If the formalization succeeds, the central forcing symmetry argument is verified; if it fails, the failure will pinpoint the exact step where the informal proof is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"As a non-finding: I carefully read the central argument and did not identify a load-bearing error. Proposition 26 is the crux: it assumes, for every n, alpha, gamma, that ˙En,alpha and ˙B_gamma are noncompact projections and ˙En,gamma(n) ≤K ˙B_gamma, and then builds two disjoint gamma1, gamma2 with ˙B_gamma1 ˙B_gamma2 non-compact. The reader's weakest assumption was the symmetry/Δ-system core of Lemma 25; I checked how that lemma is applied and it appears sound, so I do not see a genuine gap. The main reason for my non-finding is not that the argument is obviously valid, but that the proof's leverage comes from a standard combination of forcing symmetry, nice names, and the Δ-system lemma; the steps that could have failed (support-countability, disjointness via Lemma 15, symmetry via Lemma 19) are explicitly addressed. One small point that deserves attention is the role of CH in Lemma 25: the proof uses a 'low-level' argument to identify names after mapping S_alpha to N, and it is not machine-checked. But this is a verification gap, not a discovered bug. Hence I find no concrete attack that would move the verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a consistent non-embeddability theorem for the Calkin algebra: in the Cohen model, obtained by adding ω2 Cohen reals over a model of CH, the abelian C*-algebra (c0(2^ω))^N does not embed into Q(ℓ2). The central technical result is Proposition 26, a statement about P-names under CH: for families of names for noncompact projections E_{n,α} and B_γ in B(ℓ2) with E_{n,γ(n)} ≤_K B_γ, there are disjoint γ1,γ2 ∈ ωN_2 such that B_{γ1}B_{γ2} is forced to be noncompact. The proof uses Δ-system arguments, nice-name symmetry, and a carefully chosen pair ξ,η that avoids the supports of auxiliary functions. Theorem 27 then applies Proposition 26 to an embedding in the Cohen model, and the final section derives consequences for reduced products and coronas of stabilizations.","tokens_in":17005,"tokens_out":26577,"duration_ms":241640,"significance":"If the argument is correct, this is a valuable contribution: it provides a new consistent example of a C*-algebra of density continuum that fails to embed into the Calkin algebra, and it does so in the Cohen model, where the quotient-rigidity phenomena used by earlier non-embeddability results (OCA/PFA) are absent. The proof is self-contained and uses standard forcing technology without fitted parameters; the main forcing lemma is stated in a checkable form. The applications to reduced products and coronas are natural and widen the scope of the theorem. The main verification risk, the symmetry/Δ-system core of Lemma 25, is addressed explicitly, and I did not find a concrete gap.","major_comments":[],"minor_comments":[{"comment":"Equation (25) states only ||E(v)|| ≥ 0.99, but Lemma 11 requires the strict inequality > 99/100. Since l>99 already gives 1 - 1/(l+1) > 99/100, the proof should either state the strict bound or choose l so that the strict inequality is explicit.","section":"§3, Proposition 26 (around Eq. (25))"},{"comment":"The proof says the family (S_{n,α}\\Δ) consists of non-empty pairwise disjoint sets; if some S_{n,α}\\Δ is empty, this is not literally true. The argument still works if empty members are ignored, but a clarifying sentence should be added.","section":"§3, Lemma 15"},{"comment":"The reduction 'without loss of generality we may assume that Δ and S_α\\Δ are infinite' is not justified in the text. If Δ is finite or some S_α\\Δ is finite, the argument can be made to work by reserving a finite subset of N for the S_α\\Δ part, but this should be explained.","section":"§3, Lemma 25"},{"comment":"The proof works in the generic extension and then cites Proposition 26, which is a theorem about ground-model names. The paper should state that one applies Proposition 26 to names for T(χγ) with γ in the ground model; a disjoint pair of ground-model functions already contradicts injectivity.","section":"§3, Theorem 27"},{"comment":"The notation 'I={{0}}' is presumably meant to be 'I={∅}', the trivial ideal. If the paper's convention for an ideal on N requires containing all finite sets, the trivial ideal should be defined explicitly or replaced by an ideal with infinite quotient.","section":"§4, Corollary 29"},{"comment":"The corollary uses the fact that Q(ℓ2) admits a pairwise orthogonal family of 2^ω projections. This should be stated explicitly, for example via an almost disjoint family of subsets of N and the embedding of ℓ∞/c0 into Q(ℓ2).","section":"§5, Corollary 31"}],"recommendation":"minor_revision","confidential_remarks":"The paper is technically sound in my reading, and the central forcing argument is original and clearly presented modulo local clarifications. The minor issues listed above are presentation-level and do not affect the main theorem. A machine-check of Proposition 26 would be a useful verification step, but I found no concrete error. Recommendation: minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real result. In the Cohen model, (c0(2^ω))^N does not embed into Q(ℓ2). Prior examples of non-embeddable abelian algebras of density continuum lived under OCA/PFA and used rigidity of quotients; this one works where rigidity fails, using Cohen forcing and symmetric names. That is a genuinely different technique, and the consequences for reduced products and coronas are honest corollaries, not afterthoughts.\n\nWhat's good: the proof is detailed and self-contained. The key Proposition 26 is the core. The setup with nice names for very orthonormal sequences, the Δ-system argument, and the symmetry extension Lemma 19 all line up. I read through the part where they choose ξ, η with disjoint supports (Lemma 14), then use Lemma 19 to build symmetric conditions extending p while deciding h_ξ(n) and h_η(n), then pick v forced to equal both v_{n,γξ(n)}(l) and its σ-image. The inequality chain at the end contradicts (29) exactly. The projection-poset step in Theorem 27 is standard. There is no circularity: the non-embeddability is derived in ZFC plus the Cohen model assumptions, and the citations to Dow and to Brech-Koszmider are contextual.\n\nSoft spots: it is a hard proof to verify rapidly. The place where a bug would most likely hide is Lemma 25: you need to identify, after mapping each S_α to N, countably many sequences of names and get exact equality of the translated names on a size-ω2 subfamily. The authors point out that CH gives only ω1 many possibilities, so they get a large subfamily with identical translated names. That is fine, but the identification of names is a 'low-level' syntactic argument that is not machine-checked. The same is true of the whole forcing symmetry argument. So the main residual risk is a subtle error in name manipulation, not a conceptual one. Also, Proposition 26 assumes CH in the ground model; they note the proof works for adding more Cohen reals, but the written result is for ω2, which matches the density of the algebra. That is fine.\n\nWho this is for: anyone working on set-theoretic operator algebras, especially the class E and quotients of B(ℓ2). A serious referee should engage with it. The paper deserves peer review; if I were editor I would send it out. My own verdict is that the result is very likely correct; I just have not machine-checked it.","headline":"A new and believable Cohen-model counterexample to C*-algebra embeddability into the Calkin algebra; the proof is intricate but I couldn't find a hole.","tokens_in":17416,"tokens_out":2283,"would_cite":true,"duration_ms":22687,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L05","03E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"In the Cohen model, the abelian C*-algebra $(c_0(2^\\omega))^\\mathbb{N}$ cannot be embedded into the Calkin algebra $\\mathcal{Q}(\\ell_2)$, a new consistent counterexample of density continuum.","keywords":["Calkin algebra","Cohen model","C*-algebra embeddings","reduced products","corona algebras","forcing","abelian C*-algebras"],"falsifier":"To test the theorem, one can look inside the Cohen model for an injective *-homomorphism from $(c_0(2^\\omega))^\\mathbb{N}$ into $\\mathcal{Q}(\\ell_2)$; Proposition 26 forces any two disjoint coordinate sequences $\\gamma_1,\\gamma_2$ to have non-compact product of their lifted projections. A concrete counterexample would be an embedding for which two such products are compact, and a concrete check of the proof's vulnerability is whether every nice name for a vector in $\\ell_2$ in this forcing indeed has countable support and whether the support family can be thinned to a $\\Delta$-system with a countable root.","tokens_in":16461,"feed_emoji":"🧮","tokens_out":17132,"duration_ms":130093,"temperature":0.7,"pith_summary":"The Calkin algebra $\\mathcal{Q}(\\ell_2)$ is the quotient of all bounded operators on a separable Hilbert space by the compact operators, and the paper asks which C*-algebras embed into it. It proves that in the Cohen model, obtained by adding $\\omega_2$ Cohen reals to a model of the continuum hypothesis, the countable product $(c_0(2^\\omega))^\\mathbb{N}$ of copies of the abelian algebra $c_0(2^\\omega)$ (functions on a set of size continuum vanishing at infinity) does not embed into $\\mathcal{Q}(\\ell_2)$, although each single copy does embed. The significance is that this is a concrete abelian C*-algebra of density continuum that consistently sits outside the class of embeddable algebras, and the proof uses symmetry of the Cohen forcing rather than the rigidity of quotient structures used by earlier examples. From the main theorem the paper derives non-embeddability of reduced products and of coronas of stabilizations of algebras with large families of orthogonal projections.","feed_headline":"In the Cohen model, no (c0(2^ω))^N embeds into Calkin algebra","feed_subtitle":"An abelian C*-algebra of density continuum that resists the Calkin algebra, even though every single copy embeds.","key_machinery":"The load-bearing mechanism is a Cohen-forcing symmetry lemma, Lemma 25 together with Proposition 26. Given any family of countable supports $S_\\alpha$ forming a $\\Delta$-system with root $\\Delta$, and nice names $\\dot v_\\alpha(l)$ for the very orthonormal vectors produced by Lemma 10, the lemma finds a large subfamily and involutive permutations $\\sigma_{\\alpha,\\beta}$ of $\\omega_2$ that swap $S_\\alpha$ with $S_\\beta$, fix $\\Delta$, and send the name $\\dot v_\\alpha(l)$ exactly to $\\dot v_\\beta(l)$ for every $l$. Proposition 26 then combines these permutations with the fact that compact operators are asymptotically small on very orthonormal sequences: if two embedded projections $E_{n,\\gamma_\\xi(n)}$ and $E_{n,\\gamma_\\eta(n)}$ both lie below $B_{\\gamma_\\xi}$ and $B_{\\gamma_\\eta}$, the symmetry gives a single vector on which both projections have norm near 1, so their inner product exceeds 1/2, while the compactness of $B_{\\gamma_1}B_{\\gamma_2}$ forces the same inner product below 1/2. That contradiction proves the two $B$'s cannot have compact product.","core_discovery":"The paper's central claim is Theorem 27: assuming $\\mathsf{CH}$ and forcing with finite partial functions from $\\omega_2$ to $\\{0,1\\}$ (so that $2^\\omega=\\omega_2$ in the extension), there is no injective *-homomorphism from $(c_0(2^\\omega))^\\mathbb{N}$ into $\\mathcal{Q}(\\ell_2)$. The proof supposes such an embedding $T$ exists and considers the characteristic projections $\\chi_{n,\\alpha}$ and $\\chi_\\gamma$ in the product algebra. After lifting their images to projections $E_{n,\\alpha}$, $B_\\gamma$ in $\\mathcal{B}(\\ell_2)$, the algebra ordering gives $E_{n,\\gamma(n)}\\leq_K B_\\gamma$ whenever $\\gamma(n)=\\alpha$. A combinatorial symmetry argument with nice names and $\\Delta$-systems produces two disjoint functions $\\gamma_1,\\gamma_2$ with no coordinate agreement; the same symmetry forces the product $B_{\\gamma_1}B_{\\gamma_2}$ to be non-compact, contradicting $T(\\chi_{\\gamma_1}\\chi_{\\gamma_2})=0$. The contradiction does not use any rigidity of the Calkin algebra, so the example works in a model where quotient rigidity is known to fail.","pith_inferences":["The support-symmetry technique is probably adaptable to other quotient algebras where projections lift, such as $\\ell_\\infty/c_0$ or coronas of other stabilizations, yielding Cohen-model non-embeddings of analogous product algebras.","The proof uses only countably many $\\Delta$-systems with a common countable root, so the same method may work under hypotheses weaker than full $\\mathsf{CH}$, provided the relevant forcing names still have countable supports.","The paper leaves open whether the density of a non-embeddable algebra in the Cohen model can be lowered below the continuum; a natural test is whether a product of $\\omega_1$ copies or another smaller abelian algebra can also be forced to stay outside the class."],"forward_implications":["In the Cohen model, no product $\\prod_{n\\in\\mathbb{N}} A_n$ embeds into $\\mathcal{Q}(\\ell_2)$ when every $A_n$ has a pairwise orthogonal family of $2^\\omega$ projections (Corollary 29).","The same conclusion holds for reduced products $\\prod_n A_n / \\bigoplus_I A_n$ whenever $\\mathcal{P}(\\mathbb{N})/I$ is infinite and each $A_n$ has such a projection family (Theorem 28).","Coronas $\\mathcal{Q}(A\\otimes\\mathcal{K}(\\ell_2))$ of stabilizations of such algebras also fail to embed into the Calkin algebra in the Cohen model; in particular $\\mathcal{Q}(\\mathcal{Q}(\\ell_2)\\otimes\\mathcal{K}(\\ell_2))$ does not embed (Corollary 31).","Under $\\mathsf{CH}$, the companion universality result implies that $(c_0(2^\\omega))^\\mathbb{N}$ does embed into the Calkin algebra, so embeddability of this concrete algebra is independent of ZFC.","The non-embeddability persists in a model where rigidity of quotients fails, so the obstruction is not a by-product of trivial automorphism structure; it is a separate phenomenon."],"supporting_citations":[{"why":"Supplies the Δ-system theorem for countable families under CH and the forcing-automorphism lemma that lets the proof transfer names by permutations.","marker":"[22]"},{"why":"Supplies the lifting result used to pass from projections in the Calkin algebra to projections in B(ℓ2), and the general framework for products, reduced products, and coronas.","marker":"[15]"},{"why":"Gives the companion positive theorem that under CH all C*-algebras of density ω1 embed, so the new non-embeddability result is a consistency result rather than a ZFC fact.","marker":"[16]"},{"why":"Provides earlier consistent non-embeddability results based on trivial endomorphisms that the present argument avoids; the paper compares its product examples with the ones from this source.","marker":"[34]"},{"why":"Supplies the almost-disjoint family argument showing that c0(2^ω) itself embeds into ℓ∞/c0, the baseline fact that the failure is specific to the countable product.","marker":"[4]"},{"why":"One of the prior sources of consistent non-embeddable abelian algebras and reduced products that the paper enlarges with the Cohen-model examples.","marker":"[35]"}],"fun_headline_variants":["Cohen model blocks product of c0(2^ω) from Calkin algebra","No product of continuum c0's into Calkin algebra in Cohen model","Calkin algebra immune to infinite product of c0 in Cohen model","Consistent failure: product of c0(2^ω) avoids Calkin embedding","Product of c0(2^ω) fails to embed into Calkin in Cohen model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction leans on the assumption that the continuum hypothesis holds in the ground model and that the forcing names for the vectors have countable supports; if any relevant name had uncountable support, the symmetry step that identifies the two sides of the contradiction would fail.","fun_headline_variants_meta":{"raw":{"variants":["Cohen model blocks product of c0(2^ω) from Calkin algebra","No product of continuum c0's into Calkin algebra in Cohen model","Calkin algebra immune to infinite product of c0 in Cohen model","Consistent failure: product of c0(2^ω) avoids Calkin embedding","Product of c0(2^ω) fails to embed into Calkin in Cohen model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000774,"raw_usage":{"total_tokens":3511,"prompt_tokens":1119,"completion_tokens":2392,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":735,"completion_tokens_details":{"reasoning_tokens":2283}},"tokens_in":735,"tokens_out":2392,"duration_ms":14738,"temperature":1.0,"reasoning_tokens":2283,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:11:00.433969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the theorem, one can look inside the Cohen model for an injective *-homomorphism from $(c_0(2^\\omega))^\\mathbb{N}$ into $\\mathcal{Q}(\\ell_2)$; Proposition 26 forces any two disjoint coordinate sequences $\\gamma_1,\\gamma_2$ to have non-compact product of their lifted projections. A concrete counterexample would be an embedding for which two such products are compact, and a concrete check of the proof's vulnerability is whether every nice name for a vector in $\\ell_2$ in this forcing indeed has countable support and whether the support family can be thinned to a $\\Delta$-system with a countable root.","supporting_citations":[{"cited_title":"Jech, Set theory , Springer Monographs in Mathematics, Springer-Verlag, Be rlin, 2003, The third millennium edition, revised and expanded","cited_arxiv_id":null,"evidence_quote":"Supplies the Δ-system theorem for countable families under CH and the forcing-automorphism lemma that lets the proof transfer names by permutations."},{"cited_title":"Farah, Combinatorial set theory of C*-algebras , Springer Monographs in Mathematics, Springer, Cham, 2019","cited_arxiv_id":null,"evidence_quote":"Supplies the lifting result used to pass from projections in the Calkin algebra to projections in B(ℓ2), and the general framework for products, reduced products, and coronas."},{"cited_title":"Farah, I","cited_arxiv_id":null,"evidence_quote":"Gives the companion positive theorem that under CH all C*-algebras of density ω1 embed, so the new non-embeddability result is a consistency result rather than a ZFC fact."},{"cited_title":"Vaccaro, Trivial endomorphisms of the Calkin algebra , Israel J","cited_arxiv_id":null,"evidence_quote":"Provides earlier consistent non-embeddability results based on trivial endomorphisms that the present argument avoids; the paper compares its product examples with the ones from this source."},{"cited_title":"Brech, P","cited_arxiv_id":null,"evidence_quote":"Supplies the almost-disjoint family argument showing that c0(2^ω) itself embeds into ℓ∞/c0, the baseline fact that the failure is specific to the countable product."},{"cited_title":"Vignati, Logic and C*-algebras: Set theoretical dichotomies in the t heory of continuous quotients, Ph.D","cited_arxiv_id":null,"evidence_quote":"One of the prior sources of consistent non-embeddable abelian algebras and reduced products that the paper enlarges with the Cohen-model examples."}],"review_version":1}