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Every finitely generated C*-algebra without the local lifting property hides a three-dimensional operator system that fails the lifting property and yields a Smith-Ward counterexample.

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2026-07-14 07:43 UTC pith:IKQ5JIYM

load-bearing objection Clean, systematic generalization of Scherer's counterexample that produces non-exact 3D systems and a 3D nuclearity detector for every finitely generated non-LLP algebra.

arxiv 2607.11001 v1 pith:IKQ5JIYM submitted 2026-07-13 math.OA math.FA

Ubiquity of counterexamples to the Smith-Ward problem

Samuel J. Harris This is my paper
classification math.OA math.FA MSC 46L0747A1246L05
keywords Smith-Ward problemlifting propertylocal lifting propertyhyperrigidityoperator systemsmatrix rangesnuclearity detectorexactness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The Smith-Ward problem asks whether every operator on Hilbert space can be compactly perturbed so that all of its matrix ranges match those of its image in the Calkin algebra. A recent counterexample produced one three-dimensional operator system without the lifting property, but that system remained exact. This paper shows that such counterexamples are ubiquitous: for every finitely generated unital C*-algebra A that lacks the local lifting property, matrices over A contain a three-dimensional hyperrigid operator system that itself lacks the lifting property. Starting from any finite-dimensional operator system without the lifting property, the same construction inside matrices over its universal C*-algebra produces a three-dimensional system that fails both the lifting property and exactness. A special case supplies a three-dimensional nuclearity detector for unital C*-algebras. The result therefore converts every known failure of the local lifting property into a concrete Smith-Ward counterexample of the lowest possible dimension that can fail.

Core claim

Whenever A is a finitely generated unital C*-algebra without the local lifting property, there exists n and a three-dimensional hyperrigid operator system S inside M_{n+2}(A) that fails the lifting property; moreover, if T is any finite-dimensional operator system without the lifting property, then M_{n+2}(C_u^*(T)) contains such an S that also fails exactness, and a special case of this construction yields a three-dimensional nuclearity detector.

What carries the argument

An explicit three-dimensional operator system S = span{I, D, G} inside M_{n+2}(A), where D is the diagonal matrix with entries 1,…,n+2 and G is a self-adjoint matrix whose off-diagonal blocks encode a generating set of unitaries of A; hyperrigidity of S is proved by showing that any unital completely positive extension of an injective representation must reduce the Stinespring space and therefore coincide with the original representation.

Load-bearing premise

The transfer of failure of exactness or the local lifting property from the ambient algebra down to the three-dimensional subsystem rests on the claim that hyperrigidity is preserved under the left-injective and right-injective operator-system tensor products.

What would settle it

Exhibit a finitely generated unital C*-algebra A without the local lifting property for which every three-dimensional operator subsystem of every matrix algebra over A still possesses the lifting property, or show that hyperrigidity fails to pass to the el or er tensor product with some unital C*-algebra.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every finitely generated C*-algebra known to lack the local lifting property immediately supplies a concrete three-dimensional Smith-Ward counterexample.
  • There exist three-dimensional operator systems that simultaneously fail the lifting property and exactness, unlike the original counterexample which remains exact.
  • A three-dimensional nuclearity detector exists that is not a subsystem of any finite-dimensional C*-algebra.
  • Anderson’s 1978 example of a finitely generated C*-algebra with Ext not a group already yields a Smith-Ward counterexample of dimension three inside matrices of size at most five.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same matrix construction may produce three-dimensional systems that detect other approximation properties (for example the weak expectation property) once suitable duals are examined.
  • If the dual of the constructed nuclearity detector turns out to be a WEP detector, then both the Smith-Ward problem and Connes’ embedding problem would be encoded by the uniqueness of tensor products of a single three-dimensional operator system and its dual.
  • The bound n 一 2(dim T - 1) suggests that the matrix size needed for a three-dimensional counterexample grows only linearly with the dimension of the starting non-lifting system.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper constructs, for every finitely generated unital C*-algebra A generated by n unitaries, an explicit three-dimensional operator system S = span{I, D, G} inside M_{n+2}(A) that is hyperrigid (Theorem 2.4). The matrices D (diagonal with entries 1,...,n+2) and G (generators of A placed in the first row/column together with a path of 1's) are defined so that the spectral projections of D and the partial-isometry blocks of G force any ucp extension of an injective *-homomorphism to coincide with the homomorphism itself via a Stinespring argument. This yields, for every finitely generated A without the LLP, a three-dimensional hyperrigid S subset M_{n+2}(A) that fails the LP (Corollary 3.5), and, when T is any finite-dimensional operator system without the LP, a three-dimensional S inside M_{n+2}(C_u^*(T)) that fails both LP and exactness (Theorem 3.7). A special case produces a three-dimensional nuclearity detector (Theorem 4.2).

Significance. The work substantially generalizes Scherer's recent negative solution of the Smith-Ward problem by removing the dependence on Ext(A) failing to be a group and by producing counterexamples that simultaneously fail exactness. The same construction supplies the first three-dimensional nuclearity detector, improving on Kavruk's four-dimensional example. The proofs are self-contained once standard operator-system facts are granted; the matrices are given explicitly and the Stinespring analysis is written out in full. These results demonstrate that three-dimensional operator systems are already rich enough to encode both the failure of the Smith-Ward property and the detection of nuclearity.

minor comments (4)
  1. In the definition of G (display (2) and the subsequent matrix), the index on the unitaries is written u_i while the sum runs over j; a uniform index would improve readability.
  2. Proposition 3.2 is elementary and could be shortened or moved to a remark; the citation to Blecher is already acknowledged.
  3. The bound n ≤ 6 in Theorem 4.2 is correct but the paper never exhibits an explicit generating set of unitaries for C_u^*(W_{3,2}); a short remark that such a set exists would make the bound fully constructive.
  4. A few minor typos appear (e.g., "Smith-W ARD" in the running header, "e j,j+1" missing the tensor symbol in places). These are easily corrected.

Circularity Check

1 steps flagged

Self-contained Stinespring construction of hyperrigid 3D systems; only minor non-load-bearing self-citations to prior tensor-product facts

specific steps
  1. self citation load bearing [Prop. 3.3 and Thm. 4.2 (citing [9, Thm. 3.10] and [9, Prop. 6.2])]
    "C^*_env(S ⊗_el B)=A ⊗_el B o o by [9, Theorem 3.10] o o S ⊗_min A ⊆ M_{n+2}(C_u^*(W_{3,2})) ⊗_min A is hyperrigid o o by the proof of [9, Proposition 6.2]"

    These two citations supply the C*-envelope identification for hyperrigid systems and the preservation of hyperrigidity under min/ess tensors. They are not load-bearing for the existence or non-LP of the three-dimensional S itself (which is proved directly in §2), but they are the only self-references that appear in the transfer arguments of §3–4.

full rationale

The core derivation (explicit matrices D,G from unitary generators of A, Lemmas 2.2–2.3 on Stinespring subspaces, Theorem 2.4 uniqueness of ucp extensions) is fully written out from first principles (Arveson, Stinespring, functional calculus) with no parameters fitted to data and no reduction of the target claim to its own definition. Propositions 3.3–3.4 transfer hyperrigidity and exactness/LLP under el/er tensors by a short multiplicative-domain argument that does not presuppose the non-LP conclusion. Applications (Cor. 3.5, Thm. 3.7, Thm. 4.2) then follow by contraposition and the universal property of C_u^*(T), using only standard external facts (Kavruk’s W_{3,2} detector, non-exactness of C_u^*(C_3)). The two self-citations to the author’s earlier papers supply routine envelope and min-tensor hyperrigidity lemmas already proved elsewhere; they are not used to force uniqueness or to define the objects under study. No self-definitional loop, fitted-input-as-prediction, or ansatz-smuggling occurs. Score 1 reflects only the presence of those minor supporting self-cites.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The paper works entirely inside the standard framework of unital operator systems, C*-algebras, completely positive maps, Stinespring dilations, and the five operator-system tensor products of Kavruk–Paulsen–Todorov–Tomforde. No free parameters are fitted. The only non-standard ingredients are the concrete matrices D and G (ad-hoc but fully explicit) and the appeal to known non-LLP/non-exact examples (Anderson, Haagerup–Thorbjørnsen, Kirchberg–Wassermann, Kavruk).

axioms (5)
  • standard math Arveson’s extension theorem and the Choi–Effros representation of operator systems by matrix ranges
    Used throughout to identify the LP with the Smith-Ward property and to extend ucp maps.
  • domain assumption Existence of finitely generated unital C*-algebras without LLP (Anderson 1978; Haagerup–Thorbjørnsen for C_r^*(F_2))
    Supplies the input algebras to which Theorem 2.4 and Corollary 3.5 are applied.
  • domain assumption C_u^*(T) fails exactness whenever dim T ≥ 3 (Kirchberg–Wassermann via Kavruk Prop. 6.13)
    Used in Theorem 3.7 to conclude that the constructed three-dimensional S is non-exact.
  • domain assumption W_{3,2} is a nuclearity detector (Kavruk 2015)
    Lifted to C_u^*(W_{3,2}) in Proposition 4.1 and then to the three-dimensional S in Theorem 4.2.
  • ad hoc to paper Hyperrigidity is preserved under the el and er operator-system tensor products (Proposition 3.3)
    Proved in the paper but is the key new technical lemma that transfers exactness/LLP from S to A.
invented entities (1)
  • The concrete matrices D (diagonal with entries 1,…,n+2) and G (generators of A placed in the first row/column together with a path of 1’s) no independent evidence
    purpose: Generate a three-dimensional hyperrigid operator system inside M_{n+2}(A)
    Explicitly defined in equations (1)–(2); no independent existence claim beyond the construction.

pith-pipeline@v1.1.0-grok45 · 21533 in / 3206 out tokens · 23899 ms · 2026-07-14T07:43:39.765980+00:00 · methodology

0 comments
read the original abstract

The Smith-Ward problem about matrix ranges, posed in the 1980s, was recently resolved in the negative by Marcel Scherer (arXiv:2607.04274) by obtaining a three-dimensional operator system $\mathcal{S} \subseteq M_4(C_r^*(\mathbb{F}_2))$ without the lifting property. However, this operator system must be exact. In this paper we show that, for every finitely generated $C^*$-algebra $\mathcal{A}$ without the local lifting property (LLP), there exists a three-dimensional operator system $\mathcal{S} \subseteq M_{n+2}(\mathcal{A})$ without the lifting property (LP), thus generalizing Scherer's result and eliminating the reliance on $\text{Ext}(\mathcal{A})$ not being a group. In particular, we prove that whenever $\mathcal{T}$ is a finite-dimensional operator system without the LP, then $M_{n+2}(C_u^*(\mathcal{T}))$ contains a $3$-dimensional operator system without the LP for some $n \leq 2(\dim(\mathcal{T})-1)$. In this way, we yield a plethora of counterexamples to the Smith-Ward problem. In particular, unlike Scherer's example, these three-dimensional operator systems fail both the LP and exactness. We also prove the existence of a three-dimensional operator system that detects nuclearity for unital $C^*$-algebras, strengthening previous work of Kavruk (J. Funct. Anal., volume 269, 2015).

discussion (0)

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Reference graph

Works this paper leans on

20 extracted references · 3 linked inside Pith

  1. [1]

    Anderson,AC ∗-algebra for whichExt(A)is not a group, Annals of Mathematics107(1978), 455–458

    J. Anderson,AC ∗-algebra for whichExt(A)is not a group, Annals of Mathematics107(1978), 455–458

    1978

  2. [2]

    Arveson,Subalgebras ofC ∗-algebras, Acta Mathematica123(1969), 141–224

    W. Arveson,Subalgebras ofC ∗-algebras, Acta Mathematica123(1969), 141–224

    1969

  3. [3]

    2, 329–355

    ,Notes on extensions ofC ∗-algebras, Duke Journal of Mathematics44(1977), no. 2, 329–355

    1977

  4. [4]

    ,The noncommutative Choquet boundary II: Hyperrigidity, Israel Journal of Mathematics184 (2011), 349–385

    2011

  5. [5]

    Choi,A SimpleC ∗-algebra generated by two finite-order unitaries, Canadian Journal of Mathe- matics31(1979), no

    M.-D. Choi,A SimpleC ∗-algebra generated by two finite-order unitaries, Canadian Journal of Mathe- matics31(1979), no. 4, 867–880

    1979

  6. [6]

    Choi and E.G

    M.-D. Choi and E.G. Effros,Injectivity and Operator Spaces, Journal of Functional Analysis24(1977), 156–209

    1977

  7. [7]

    Haagerup and S

    U. Haagerup and S. Thorbjørnsen,A new application of random matrices: Ext(C ∗ red(F2))is not a group, Annals of Mathematics162(2005), 711–775

    2005

  8. [8]

    Harris,Four-dimensional operator systems without the lifting property, preprint (arXiv:2508.00113) (2025)

    S.J. Harris,Four-dimensional operator systems without the lifting property, preprint (arXiv:2508.00113) (2025). UBIQUITY OF COUNTEREXAMPLES TO THE SMITH-W ARD PROBLEM 15

    Pith/arXiv arXiv 2025

  9. [9]

    Harris and S.-J

    S.J. Harris and S.-J. Kim,Crossed products of operator systems, Journal of Functional Analysis276 (2019), no. 7, 2156–2193

    2019

  10. [10]

    Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen,MIP ∗ = RE, preprint (arXiv:2001.04383 [math- ph]) (2020)

    Pith/arXiv arXiv 2001

  11. [11]

    Kavruk, V.I

    A.S. Kavruk, V.I. Paulsen, I.G. Todorov, and M. Tomforde,Tensor products of operator systems, Journal of Functional Analysis261(2011), no. 2, 267–299

    2011

  12. [12]

    ,Quotients, exactness and nuclearity in the operator system category, Advances in Mathematics 235(2013), 321–360

    2013

  13. [13]

    Kavruk,Nuclearity related properties in operator systems, Journal of Operator Theory71(2014), no

    A.S. Kavruk,Nuclearity related properties in operator systems, Journal of Operator Theory71(2014), no. 1, 95–156

    2014

  14. [14]

    ,On a non-commutative analogue of a classical result of Namioka and Phelps, Journal of Func- tional Analysis269(2015), 3282–3303

    2015

  15. [15]

    Kirchberg,On non-semisplit extensions, tensor products and exactness of groupC ∗-algebras, Inven- tiones mathematicae112(1993), no

    E. Kirchberg,On non-semisplit extensions, tensor products and exactness of groupC ∗-algebras, Inven- tiones mathematicae112(1993), no. 3, 449–490

    1993

  16. [16]

    Kirchberg and S

    E. Kirchberg and S. Wassermann,C ∗-algebras generated by operator systems, Journal of Functional Analysis155(1998), 324–351

    1998

  17. [17]

    Paulsen, I.G

    V.I. Paulsen, I.G. Todorov, and M. Tomforde,Operator system structures on ordered spaces, Proceedings of the London Mathematical Society102(2011), no. 1, 25–49

    2011

  18. [18]

    Scherer,A three-dimensional operator system without the Smith-Ward property, preprint (arXiv:2607.04274) (2026)

    M. Scherer,A three-dimensional operator system without the Smith-Ward property, preprint (arXiv:2607.04274) (2026)

    Pith/arXiv arXiv 2026

  19. [19]

    Smith and J.D

    R.R. Smith and J.D. Ward,Matrix ranges for Hilbert space operators, American Journal of Mathematics 102(1980), 1031–1081

    1980

  20. [20]

    Wassermann,ExactC ∗-algebras and related topics, Lecture notes series at Seoul National University, vol

    S. Wassermann,ExactC ∗-algebras and related topics, Lecture notes series at Seoul National University, vol. 19, Seoul National University, 1994. Northern Arizona University, Department of Mathematics & Statistics, 801 S. Os- borne Dr., Flagstaff, AZ, 86011 USA Email address:samuel.harris@nau.edu

    1994