Products of C*-algebras that do not embed into the Calkin algebra

Damian G{\l}odkowski; Piotr Koszmider

Products of C*-algebras that do not embed into the Calkin algebra

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.11191 v1 pith:4KYRFCXX submitted 2024-12-15 math.OA math.FAmath.LO

Damian G{l}odkowski , Piotr Koszmider This is my paper · ORCID

classification math.OAmath.FAmath.LO MSC 46L0503E35

keywords CalkinalgebraCohenmodelC*-algebraembeddingsreducedproductscoronaalgebrasforcingabelianC*-algebras

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What'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.

Soft 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.

Who 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.

Referee Report

0 major / 6 minor

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.

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.

minor comments (6)

[§3, Proposition 26 (around Eq. (25))] 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.
[§3, Lemma 15] 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.
[§3, Lemma 25] 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.
[§3, Theorem 27] 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.
[§4, Corollary 29] 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.
[§5, Corollary 31] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-embeddability proof is self-contained and does not reuse its target as an input.

full rationale

The paper's central derivation is self-contained within ZFC plus the Cohen model assumptions. Proposition 26 assumes the existence of projection names E_{n,alpha} and B_gamma satisfying E_{n,gamma(n)} <=K B_gamma and derives, via Lemma 25's Delta-system and forcing-symmetry argument, two disjoint gamma1 and gamma2 with B_{gamma1}B_{gamma2} non-compact; Theorem 27 then contradicts the multiplicative structure of a hypothetical embedding. No parameter is fitted to data, and no instance of the target proposition, namely the non-embeddability of (c0(2^omega))^N into the Calkin algebra, is used as an input. Citations to prior work are either standard references, such as Jech for the Delta-system lemma and forcing facts, or contextual comparisons with results by Vaccaro, McKenney and Vignati, and Farah, Hirshberg and Vignati, and are not load-bearing for the main proof. The use of CH is part of the Cohen model hypothesis rather than a circular assumption. The one substantive unverified step is the forcing-symmetry identification in Lemma 25, but that is a potential verification gap, not a circular reduction of the theorem to its own conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; this is a pure consistency proof. The assumptions are the standard axioms of set theory, the ground model CH used to define the Cohen model, and standard theorems about C*-algebras such as the lifting of projections from coronas. No new entities are postulated.

assumptions (4)

standard math ZFC consistency and forcing machinery for adding ω2 Cohen reals to a model of CH
The entire proof is carried out in a forcing extension of a model of CH. This is the standard framework for set-theoretic independence results.
standard math The C*-algebra lifting theorem [15, Lemma 3.1.13], which lifts projections from the Calkin algebra to B(ℓ2)
Used in Theorem 27 to obtain projections E_{n,α} and B_γ in B(ℓ2) from the embedding. This is a standard theorem about multiplier/corona algebras.
domain assumption CH in the ground model, used to count the number of names in Lemma 25 and to apply the Δ-system lemma for ω2 (Lemma 13)
The Cohen model is defined from a model of CH; the proofs in Section 3 explicitly invoke CH.
standard math The existence of almost disjoint families of size 2^ω, used in Lemma 4 to embed c0(2^ω) into ℓ∞/c0
This is a ZFC fact, cited for comparison and used in the introduction.

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Pith. "Pith review of Products of C*-algebras that do not embed into the Calkin algebra." pith.science (2026). https://pith.science/paper/4KYRFCXX

@misc{pith2026241211191,
  author       = {Pith},
  title        = {Pith review of: Products of C*-algebras that do not embed into the Calkin algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KYRFCXX}},
  note         = {Machine review of arXiv:2412.11191}
}

abstract

We consider the Calkin algebra $\mathcal{Q}(\ell_2)$, i.e., the quotient of the algebra $\mathcal B(\ell_2)$ of all bounded linear operators on the separable Hilbert space $\ell_2$ divided by the ideal $\mathcal K(\ell_2)$ of all compact operators on $\ell_2$. We show that in the Cohen model of set theory ZFC there is no embedding of the product $(c_0(2^\omega))^{\mathbb{N}}$ of infinitely many copies of the abelian C*-algebra $c_0(2^\omega)$ into $\mathcal{Q}(\ell_2)$ (while $c_0(2^\omega)$ always embeds into $\mathcal{Q}(\ell_2)$). This enlarges the collection of the known examples due to Vaccaro and to McKenney and Vignati of abelian algebras, asymptotic sequence algebras, reduced products and coronas of stabilizations which consistently do not embed into the Calkin algebra. As in the Cohen model the rigidity of quotient structures fails in general, our methods do not rely on these rigidity phenomena as is the case of most examples mentioned above. The results should be considered in the context of the result of Farah, Hirshberg and Vignati which says that consistently all C*-algebras of density up to $2^\omega$ do embed into $\mathcal{Q}(\ell_2)$. In particular, the algebra $(c_0(2^\omega))^{\mathbb{N}}$ consistently embeds into the Calkin algebra as well.

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Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Bartoszy´ nski, H

T. Bartoszy´ nski, H. Judah,Set theory: on the structure of the real line. Wellesley, MA: A. K. Peters Ltd. (1995)

work page 1995

[2] [2]

Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol

B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer- Verlag, Berlin, 2006, Theory of C*-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III

work page 2006

[3] [3]

Brech, P

C. Brech, P. Koszmider, On universal Banach spaces of density continuum , Isr. J. Math. 190, (2012) pp. 93–110

work page 2012

[4] [4]

Brech, P

C. Brech, P. Koszmider, ℓ∞ -sums and the Banach space ℓ∞ /c0, Fund. Math. 224 (2014), no. 2, 175–185

work page 2014

[5] [5]

J. W. Calkin, Two-sided ideals and congruences in the ring of bounded oper ators in Hilbert space. Ann. Math. (2) 42, 839-873 (1941)

work page 1941

[6] [6]

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J. Conway, A course in functional analysis , second ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990

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[8] [8]

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