A non-nuclear $C^*$-algebra with the Weak Expectation Property and the Local Lifting Property

Gilles Pisier

REVIEW 3 major objections 3 minor 27 references

A non-nuclear $C^*$-algebra with the Weak Expectation Property and the Local Lifting Property

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Pith's one-line read A non-nuclear C*-algebra can still have both the weak expectation and local lifting properties.

desk verdict Important construction with a fixable but load-bearing gap: the proof never actually proves non-nuclearity, and the theorem as stated is false for trivial sequences like Z_n = 0. read the letter →

arxiv 1908.02705 v5 pith:LVU6DTBN submitted 2019-08-07 math.OA math.FA

Gilles Pisier This is my paper · ORCID

classification math.OAmath.FA MSC 46L0646L0746L09

keywords non-nuclearC*-algebraweakexpectationpropertylocalliftingConnes-Kirchbergproblemconealgebraapproximatelymultiplicativemapsoperatorspacesinductivelimits

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

This paper constructs the first example of a separable non-nuclear C*-algebra $A$ that has both the Weak Expectation Property (WEP) and the Local Lifting Property (LLP), two properties that together force a unique C*-norm on the tensor product $A \otimes A^{\mathrm{op}}$ and that had not previously been known to coexist outside nuclearity. The construction is designed so that $A$ has exactly the same finite-dimensional operator subspaces as the full C*-algebra $C^*(F_\infty)$ of the free group on infinitely many generators, the algebra whose possession of the WEP is the open Connes–Kirchberg problem. Because $C^*(F_\infty)$ is known to have the LLP, the new algebra is a test case: it realizes the WEP at the level of local operator-space structure without settling whether $C^*(F_\infty)$ itself has the WEP. The paper also proves a general theorem: from a sequence of C*-algebras in which each is a subquotient of the next, with linking maps that “almost allow liftings,” one can form a limit algebra whose identity map inherits the local tensor properties of the inclusions. If the main claim is right, WEP and LLP can coexist with non-nuclearity, and the obstruction to the Connes–Kirchberg problem is not a purely finite-dimensional phenomenon.

What carries the argument

The engine is the cone algebra $C_0(C)$ (the C*-algebra of continuous functions $f : [0,1] \to C$ with $f(0)=0$, where $C = C^*(F_\infty)$) together with the notion of an $\varepsilon$-morphism: a self-adjoint linear map $\psi : E_0 \to B_0$ that is almost contractive and almost multiplicative on a finite-dimensional self-adjoint domain. Lemma 6.1 shows that passing to cone algebras makes any quotient map $q$ almost allow liftings: the induced map $q_0 : C_0(C) \to C_0(B)$ can be locally lifted by $\varepsilon$-morphisms, via a quasicentral approximate unit of the kernel. Lemma 7.1 is the inductive step: given a finite-dimensional subspace $E$ of $C_0(C)$, it produces a larger subspace $E_1$ and an almost isometric, almost multiplicative inclusion $T : E \to E_1$ that extends any prescribed $u : S \to E$ ($S \subset \ell^n_1$) up to cb-norm $(1+\varepsilon)$, using the WEP of $C_0(B)$ to seed the extension $v$. Iterating Lemma 7.6 yields the sequence $(E_n, T_n)$, and the ambient quotient $L = \ell_\infty(C_0(C))/c_0(C_0(C))$ turns the almost-multiplicative links into an honest C*-algebra $A$.

What would settle it

Exhibit a WEP C*-algebra $B$, a finite-dimensional subspace $S \subset \ell^n_1$, and a complete contraction $u : S \to C_0(B)$ such that for some fixed $\delta > 0$ every extension $\tilde{u} : \ell^n_1 \to C_0(B)$ with $\tilde{u}|_S = u$ has $\|\tilde{u}\|_{\mathrm{cb}} \geq 1 + \delta$; this would refute the WEP of the cone algebra and invalidate Lemma 7.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 7.2: there exists a separable non-nuclear (in fact non-exact) C*-algebra $A$ with the WEP and the LLP. The proof builds $A$ inside the quotient algebra $L = \ell_\infty(C_0(C))/c_0(C_0(C))$ of the cone algebra of $C = C^*(F_\infty)$, as an increasing union of finite-dimensional self-adjoint subspaces $Y_n$ that are almost completely isometric to prescribed subspaces $E_n \subset C_0(C)$. The linking maps $T_n : E_n \to E_{n+1}$ are chosen to be simultaneously almost isometric and almost multiplicative ($\varepsilon$-morphisms), so the union becomes a C*-subalgebra while retaining the extension property that characterizes the WEP. Because $A$ has the WEP and locally embeds in $C$, Proposition 3.7 yields the LLP; and since the $E_n$ can be chosen to contain any prescribed dense family of finite-dimensional subspaces of $C_0(C)$, $A$ and $C$ are locally equivalent. The paper further shows the example can be made quasidiagonal and of similarity degree 3, and in Section 9 it abstracts the whole scheme into a general “inductive and projective limit” theorem for sequences of subquotients.

Load-bearing premise

The construction assumes, without proof in Lemma 7.1, that the cone algebra $C_0(B)$ of a WEP C*-algebra $B$ again has the WEP, so that the map $i_0u$ admits the almost-isometric extension $v$ that seeds the inductive step; it also relies on Kirchberg's lemma that any separable C*-algebra embeds in a separable WEP algebra.

Editorial extensions

If this is right

$A$ has a unique C*-norm on $A \otimes A^{\mathrm{op}}$, because by Kirchberg's tensor-product criterion WEP plus LLP implies $A \otimes_{\min} B = A \otimes_{\max} B$ for every $B$, in particular for $B = A^{\mathrm{op}}$.
$A$ and $C = C^*(F_\infty)$ are locally equivalent: every finite-dimensional operator subspace of $C$ embeds almost completely isometrically into $A$, and conversely by the LLP of $A$.
Choosing the prescribed subspaces densely, $A$ witnesses that the Connes–Kirchberg problem is not decided by any finite-dimensional obstruction: the WEP of $C^*(F_\infty)$ would follow from the WEP of $A$ only if the WEP were a purely local property, which it is not known to be.
The example can be modified to be unital, quasidiagonal, and of similarity degree 3, so the non-nuclearity is compatible with strong finite-dimensional approximation properties.
The general Theorem 9.2 supplies, for any suitable local property of an injective *-homomorphism $i : C \to B$ (with $C$ having the LLP and the quotient map $q$ almost allowing liftings), a C*-algebra $A$ for which the identity map has that property and $A$ is locally equivalent to $C$.

Reading between the lines

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

The construction suggests that the Connes–Kirchberg problem, if true, cannot be certified by any local property that survives passage to the inductive limit; one would need a global tensor-product argument that distinguishes $C$ from $A$ despite their identical finite-dimensional operator subspace structure.
One could test whether the identity map of $A$ admits the lifting property (LP), not just the LLP; if $A$ fails the LP while having the LLP, it would give a new separation of the two local lifting notions in the separable setting.
The inductive/projective limit picture in Section 9 may transfer to von Neumann algebras or exactness questions: replacing “cone algebra” by another liftable functor would yield algebras whose identities carry properties of the linking maps, possibly producing new counterexamples to permanence of nuclearity or exactness under such limits.
Since the sequence $(E_n)$ can be chosen to contain any prescribed subspaces, one could stress-test the construction by feeding in subspaces known to force large exactness constants, measuring how slowly the almost-multiplicative errors $\varepsilon_n$ must decay to keep the limit a C*-algebra.

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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

Pisier's paper is the real thing—first example of a non-nuclear C*-algebra with both WEP and LLP, answering an explicit question from Brown-Ozawa. The construction is original: a sequence of finite-dimensional subspaces with almost multiplicative linking maps, built via the cone algebra trick, and the general framework in Section 9 is a nice bonus. The estimates in Lemma 7.6 and the proof of WEP/LLP are careful and mostly check out.

But there is a real problem with the advertised headline. Theorem 7.2 states that for any sequence (Z_n) of f.d.s.a. subspaces of C0(C), the constructed A is non-nuclear. The proof never addresses non-nuclearity. It proves A is a C*-algebra with WEP and LLP, and the 'Moreover' clause, but there is no exactness constant argument anywhere. That is not a minor omission: if you take Z_n = 0, the construction gives A = 0, which is nuclear, so the statement as written is false. The intended theorem must include a hypothesis on (Z_n), e.g. that they form a d_cb-dense sequence in C0(C) (then A is locally equivalent to C0(C) and non-exactness follows). The outline in Section 4 suggests the author had this in mind, but it didn't make it into the theorem statement or proof. This has to be fixed.

There is also a smaller gap: A is defined as the union ∪Y_n, but the product estimate (7.7) only places ab in the closure. The line 'hence ab ∈ ∪Y_n' is not justified; taking A to be the closure fixes it, and the WEP/LLP arguments survive by density. Minor but should be corrected.

One other thing: Lemma 7.1 uses 'B and hence C0(B) has the WEP' without proof or citation. It's probably standard, but it deserves a reference, since the whole inductive step leans on it.

The general Theorem 9.2 is sketched—the proof that A is a C*-subalgebra is skipped, and the same closure issue applies. That is a soft spot in the second half, but the framework is clearly valuable.

Net: the core construction is sound and the result is significant. The non-nuclearity gap is fixable but load-bearing; the paper needs a serious revision before publication. I would still send it to a good referee—the ideas are important and the gaps are repairable. This deserves peer review, not desk rejection.

Referee Report

3 major / 3 minor

Summary. The paper constructs a separable C*-algebra A that is claimed to be non-nuclear while having both the Weak Expectation Property (WEP) and the Local Lifting Property (LLP). The construction proceeds inductively: starting from finite-dimensional self-adjoint subspaces Z_n of C0(C*(F_∞)), Lemma 7.1 produces almost multiplicative, almost isometric embeddings T_n : E_n → E_{n+1}; Lemma 7.6 assembles these into a sequence satisfying extension, cb-norm, and approximate multiplicativity conditions; Theorem 7.2 then realizes A inside the ultrapower L = ℓ_∞(C0(C))/c0(C0(C)) as the union of the Y_n = Qθ_n(E_n). The proof claims that A is a C*-subalgebra, has the WEP by Proposition 2.1, and has the LLP by local embeddability into C0(C) plus Proposition 3.7. A second part (Section 9) reformulates the construction in the setting of a sequence of C*-algebras for which each C_n is a subquotient of C_{n+1}, and uses this to transfer local properties of the linking maps to the identity map of the resulting algebra.

Significance. If the construction is correct, this is a significant result: it answers an implicit question of Kirchberg and gives new examples of non-nuclear C*-algebras with unique C*-norm on A ⊗ A^{op}. The strategy, combining local lifting with almost multiplicative maps and the LLPs of free group C*-algebras, is original and the estimates in Lemma 7.6 and the product estimate (7.5) are detailed and non-trivial. The paper also gives a general framework (Section 9) that may be useful for transferring other local properties. However, as written, the proof of Theorem 7.2 does not establish the title property (non-nuclearity), and the definition of A as a union rather than a closure creates a gap in the proof that A is a C*-algebra. These issues are load-bearing and require a major revision.

major comments (3)

[§7, Theorem 7.2] The theorem asserts the existence of a non-nuclear C*-algebra A, but the proof never proves non-nuclearity (or non-exactness). The proof shows that A has the WEP and the LLP and that each Z_n embeds almost isometrically into A; it does not show that C0(C), or any non-exact space, locally embeds into A. Section 4 promises that starting from a finite-dimensional space E1 with exactness constant >1 will ensure non-exactness of X, but Lemma 7.6 starts with E1 = Z1 for an arbitrary sequence (Z_n) and no exactness hypothesis is used in the proof of Theorem 7.2. Since exactness is a local property, if (Z_n) is chosen, for example, to be a d_cb-dense sequence so that A and C are locally equivalent (Remark 7.5), then A is non-exact and hence non-nuclear; but this choice is not stated in the theorem and the argument is not supplied. The non-nuclearity claim therefore needs either a suitable extra hypothesis on (Z_n) together with a proof of local equivalence, or a direct proof that the construction forces non-exactness.
[§7, after Eq. (7.7)] The proof defines A = ∪Y_n and then uses the product estimate (7.7), which gives d(ab, Y_{n+k+1}) → 0 for a,b ∈ Y_n, to conclude 'hence ab ∈ ∪Y_n'. This inference is invalid: (7.7) only places ab in the closure of ∪Y_n. Since a C*-subalgebra of L must be closed, the construction should define A as the closure of ∪Y_n. The subsequent WEP argument already uses density ('we may obviously assume by density that u(S) ⊂ ∪Y_m'), which is consistent with A being the closure. Defining A as the closure is a fixable modification, but as written the set A is not shown to be closed under products.
[§7, Theorem 7.2, 'Moreover' clause] The 'Moreover' clause states that for any n and any ε > 0 there is a subspace Z ⊂ A with d_cb(Z_n, Z) < 1+ε for an arbitrary sequence (Z_n). The proof explicitly adds a hypothesis not present in the statement: 'if we arrange the sequence (Z_n) so that each space in it is repeated infinitely many times'. For an arbitrary sequence, a fixed Z_n occurs only once, and the available estimate d_cb(Z_n, Y_n) ≤ (1+η_n)^2 = O(1+ε) is not enough to make the distortion below 1+ε for a prescribed ε when n is fixed. The theorem should either include the repetition hypothesis, or state the approximation property only for a cofinal subsequence of the Z_n.

minor comments (3)

[§7, Lemma 7.1] The sentence 'Since B and hence C0(B) has the WEP' is used to produce the extension v of i0u, but no justification is given for why the WEP passes from B to C0(B). This is true (C0(B) = C0 ⊗ B with C0 nuclear), but the implication should be stated or referenced for completeness.
[§9, Theorem 9.2] The proof says 'The proof that A = ∪Y_n is a C*-subalgebra is entirely analogous to that of Theorem 7.2 so we skip it.' Given the closure issue in Theorem 7.2, the analogous proof is not entirely trivial: the same union-versus-closure problem occurs. The theorem and its proof should define A as the closure of ∪Y_n (or state that the closure is taken) and indicate that the remaining arguments are unaffected.
[Throughout] There are several typos: 'tends pointwise to to 0' in the proof of Proposition 2.1, 'same conclusion hods' in the proof of Theorem 7.2, and 'unitizat ion' in Remark 7.3. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is an explicit inductive limit built from independent background results; the under-proved non-nuclearity and cone-WEP assertions are correctness gaps, not circular reductions.

full rationale

The paper's derivation chain is not circular. It starts from C=C*(F∞), a separable WEP algebra B containing C (via Kirchberg's lemma), and the known LLP of C, and builds an increasing sequence (En,Tn) inside C0(C) by Lemma 7.1. The WEP of C0(B) is used only to manufacture the extension v of i0u in the inductive step; the target WEP of A is never assumed. The resulting A=∪Yn is shown to be a C*-subalgebra from the almost-multiplicativity estimate (7.4)/(7.7); its WEP is then verified through the extension property in Proposition 2.1, and its LLP follows from Proposition 3.7 after establishing d_SC(A)=1 via dcb(Yn,En)≤(1+ηn)^2 and the LLP of C0(C). No parameter is fitted to data and no 'prediction' is a renamed input; the sequence (Zn) is arbitrary and feeds only the extra approximation clause, not the WEP/LLP conclusions. Self-citations such as [9, Th. 4.5] for formula (3.2), or [16] for the general strategy, are independent published results and do not assume the present theorem. Flagged for the record are two non-circular gaps: Lemma 7.1 asserts 'Since B and hence C0(B) has the WEP' without proof (a permanence fact, not a circular assumption), and the proof of Theorem 7.2 does not implement the Section 4 plan to start from E1 with exactness constant >1, leaving non-nuclearity underived; also (7.7) yields only d(ab,Y_{n+k+1})→0, so the line 'and hence ab∈∪Yn' should read that ab lies in the closure. These are correctness gaps and do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The construction leans on several established theorems from Kirchberg and Haagerup; no parameters are fitted to data. The central novelty is the inductive limit construction, not a new postulate.

assumptions (6)

domain assumption C*(F_infinity) has the Local Lifting Property.
Used in Lemma 6.2 to lift pi i|_E to a map u into ell_infty(C1) with controlled cb-norm; this is a theorem of Kirchberg cited in the introduction.
domain assumption Every separable C*-algebra embeds in a separable C*-algebra with the WEP.
Invoked at the start of Section 7 to obtain B with C subset B; cited to Kirchberg [10, Lemma 2.4].
domain assumption The WEP passes to the cone algebra C0(B).
Used without proof in Lemma 7.1 to extend i0u to v with controlled cb-norm; seems standard, but unproved in the paper.
domain assumption The pair (B(H), C*(F_infinity)) is nuclear.
Used in Proposition 3.7 to replace C tensor_max B with C tensor_min B; stated as Theorem 1.4.
domain assumption Haagerup's equivalence of dec-norm and cb-norm for maps into B(H), plus related results from [8].
Used in Proposition 3.7 to extend v to ~v and compute norms; cited to [8].
standard math The set of finite-dimensional subspaces of a separable operator space is d_cb-separable.
Used in Remark 7.5 to produce a dense sequence (Z_n).

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Pith. "Pith review of A non-nuclear $C^*$-algebra with the Weak Expectation Property and the Local Lifting Property." pith.science (2026). https://pith.science/paper/LVU6DTBN

@misc{pith2026190802705,
  author       = {Pith},
  title        = {Pith review of: A non-nuclear $C^*$-algebra with the Weak Expectation Property and the Local Lifting Property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVU6DTBN}},
  note         = {Machine review of arXiv:1908.02705}
}

abstract

We construct the first example of a $C^*$-algebra $A$ with the properties in the title. This gives a new example of non-nuclear $A$ for which there is a unique $C^*$-norm on $A \otimes A^{op}$. This example is of particular interest in connection with the Connes-Kirchberg problem, which is equivalent to the question whether $C^*({\bb F}_2)$, which is known to have the LLP, also has the WEP. Our $C^*$-algebra $A$ has the same collection of finite dimensional operator subspaces as $C^*({\bb F}_2)$ or $C^*({\bb F}_\infty)$. In addition our example can be made to be quasidiagonal and of similarity degree (or length) 3. In the second part of the paper we reformulate our construction in the more general framework of a $C^*$-algebra that can be described as the \emph{limit both inductive and projective} for a sequence of $C^*$-algebras $(C_n)$ when each $C_n$ is a \emph{subquotient} of $C_{n+1}$. We use this to show that for certain local properties of injective (non-surjective) $*$-homomorphisms, there are $C^*$-algebras for which the identity map has the same properties as the $*$-homomorphisms.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Arveson, Notes on extensions of C ∗-algebras, Duke Math

W. Arveson, Notes on extensions of C ∗-algebras, Duke Math. J. 44 (1977), 329–355

work page 1977

[2] [2]

Blackadar, Operator algebras, Theory of C ∗-algebras and von Neumann algebras , Springer-Verlag, Berlin, 2006

B. Blackadar, Operator algebras, Theory of C ∗-algebras and von Neumann algebras , Springer-Verlag, Berlin, 2006

work page 2006

[3] [3]

Blackadar and E

B. Blackadar and E. Kirchberg, Generalized inductive li mits of ﬁnite-dimensional C ∗- algebras. Math. Ann. 307 (1997), 343–380

work page 1997

[4] [4]

Bourgain and G

J. Bourgain and G. Pisier, A construction of L∞-spaces and related Banach spaces, Bol. Soc. Brasil. Mat. 14 (1983), 109–123

work page 1983

[5] [5]

Brown and N

N.P. Brown and N. Ozawa, C ∗-algebras and ﬁnite-dimensional approximations , Graduate Studies in Mathematics, 88, American Mathematical Society , Providence, RI, 2008

2008

[6] [6]

Connes and N

A. Connes and N. Higson, D´ eformations, morphismes asymptotiques et K-th´ eorie bivariante [Deformations, asymptotic morphisms and bivariant K-theo ry] C. R. Acad. Sci. Paris S´ er. I Math. 311 (1990), 101–106

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Eﬀros and Z.J

E. Eﬀros and Z.J. Ruan, Operator Spaces. Oxford Univ. Press, Oxford, 2000

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Operator algebras and their connection with Topology and Ergodic The ory

U. Haagerup, Injectivity and decomposition of complete ly bounded maps in “Operator algebras and their connection with Topology and Ergodic The ory”. Springer Lecture Notes in Math. 1132 (1985), 170–222

work page 1985