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A three-dimensional operator system in the Calkin algebra has no unital completely positive lift of the identity, giving the first three-dimensional non-exact operator system.

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2026-07-14 16:22 UTC pith:JTM7WYON

load-bearing objection Clean reduction of the Smith–Ward counterexample from four to three dimensions via an explicit hyperrigid triple; the argument holds.

arxiv 2607.04274 v2 pith:JTM7WYON submitted 2026-07-05 math.OA math.FA

A Three-Dimensional Operator System without the Smith--Ward Property

Marcel Scherer This is my paper
classification math.OA math.FA MSC 46L07
keywords Smith-Ward problemlifting propertyhyperrigidityoperator systemexactnessCalkin algebraExtmatrix range
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 paper constructs a three-dimensional operator system inside the Calkin algebra whose identity map admits no unital completely positive lift. The system is the image of a carefully chosen hyperrigid span{1, D, K} living in a matrix algebra over the reduced free-group C*-algebra on two generators; D is diagonal with four distinct eigenvalues and K is a self-adjoint matrix whose nonzero entries are the free generators and the unit. Because the system is hyperrigid, any lift of the identity would force a lift of a non-invertible Ext class, which is impossible. Equivalently, the single operator D + iK supplies a counter-example to the classical Smith–Ward problem: no compact perturbation can match its full joint matrix range. A duality theorem of Kavruk then shows that the dual of this system fails to be exact, producing the first known three-dimensional non-exact operator system. The result sharpens earlier four- and five-dimensional counter-examples and settles the dimension of the Smith–Ward obstruction at three.

Core claim

There exist self-adjoint operators D and K on a separable Hilbert space such that the three-dimensional Calkin operator system span{1, q(D), q(K)} has no unital completely positive lift of the identity; equivalently the operator T = D + iK admits no compact perturbation L with W(T + L) = W(q(T)). The dual of this system is therefore the first three-dimensional operator system that is not exact.

What carries the argument

The three-dimensional hyperrigid operator system S = span{1, D, K} inside M4(C*_r(F2)), where D is diagonal with distinct eigenvalues and the nonzero entries of K are free unitaries and the unit. Hyperrigidity (unique extension property for every unital representation) converts any supposed lift of the image of S into a lift of a non-invertible Ext class, which is forbidden.

Load-bearing premise

The existence of an injective representation of the reduced free-group C*-algebra into the Calkin algebra whose Ext class is not invertible.

What would settle it

Exhibit a unital completely positive map from the concrete three-dimensional Calkin system span{1, q(D), q(K)} into B(H) that composes with the quotient map to give the identity, or prove that every injective representation of C*_r(F2) into the Calkin algebra has invertible Ext class.

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Editorial analysis

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Referee Report

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Summary. The paper constructs a three-dimensional hyperrigid operator system S = span{1, D, K} inside M4(C*_r(F2)), with D a diagonal self-adjoint matrix having four distinct eigenvalues and K a self-adjoint matrix whose nonzero entries are the free unitaries u, v and the unit. It proves that C*(S) equals the full matrix algebra and that every unital representation of M4(C*_r(F2)) restricts to a map on S with the unique extension property (Theorem 3.2). Combining this with a non-invertible injective representation τ of C*_r(F2) into the Calkin algebra (from Haagerup–Thorbjørnsen) and the elementary Ext-obstruction of Lemma 2.1, the authors obtain a three-dimensional Calkin subsystem ẊS whose identity map admits no u.c.p. lift (Proposition 4.1). Equivalently, the operator T = T1 + i T2 formed by self-adjoint lifts of τ4(D) and τ4(K) has no compact perturbation realizing the full joint matrix range of q(T) (Theorem 4.2). By Kavruk’s duality the dual of ẊS is therefore a three-dimensional non-exact operator system (Corollary 4.3).

Significance. The result settles the generalized Smith–Ward problem in dimension three and supplies the first example of a three-dimensional operator system that fails to be exact. The reduction from Harris’ four-dimensional coding to a hyperrigid three-dimensional system is achieved by a transparent spectral-propagation argument that recovers the four projections of D from the two endpoint subspaces and the partial-isometry edges of K. All steps rely only on standard tools (Stinespring dilation, Arveson extension, Powers simplicity, Haagerup–Thorbjørnsen non-group Ext) and are written in full detail; the construction is therefore both novel and immediately usable for further work on lifting and exactness questions in low-dimensional operator systems.

minor comments (5)
  1. [Abstract] Abstract and page 1: the phrase “a counterexamples” should be “counterexamples” (or “a counterexample”).
  2. [Introduction] Page 1, line 3 of the introduction: “into the Calkin algebra gives explicit four-dimensional…” reads more smoothly as “into the Calkin algebra yields explicit four-dimensional…”.
  3. [Lemma 3.1] Lemma 3.1: the phrase “Simple computations now imply eij ⊗ 1 ∈ C*(S)” could be expanded by one sentence indicating which products of the already-obtained corners produce the remaining matrix units.
  4. [Theorem 3.2] Theorem 3.2 proof, middle of page 6: the sentence beginning “Using V(π(p4)E) ⊆ ρ(p4)L the left hand side changes to…” is slightly dense; a brief parenthetical reminder that the other summands vanish by the already-established inclusions would improve readability.
  5. [References] References: the arXiv identifier for Harris [6] is given as arXiv:2508.00113v2; if a published version appears before final production it should be updated.

Circularity Check

0 steps flagged

No significant circularity: the three-dimensional counterexample is obtained by combining an independent Ext non-group theorem with a self-contained hyperrigidity proof for a concrete operator system.

full rationale

The derivation chain is linear and non-circular. The existence of a non-invertible injective unital representation τ : C*_r(F2) o Q(H) is taken from the external Haagerup–Thorbjørnsen theorem (cited as [4]); Lemma 2.1 then shows that any u.c.p. lift would force invertibility of the Ext class, again by a standard Stinespring argument written out in full. The novel three-dimensional operator system S = span{1, D, K} is defined by concrete matrices inside M4(C*_r(F2)); Lemma 3.1 verifies that C*(S) recovers the whole matrix algebra by elementary spectral-projection and corner computations; Theorem 3.2 proves the unique-extension property (hyperrigidity) from first principles by a Stinespring dilation argument that recovers the four spectral subspaces of D from the two endpoint subspaces and propagates them along the partial-isometry edges of K. Proposition 4.1 and Theorem 4.2 simply combine these two independent ingredients: a hypothetical u.c.p. lift of id on au4(S) would extend, by hyperrigidity, to a lift of au4, contradicting non-invertibility. Corollary 4.3 invokes Kavruk’s external duality theorem. No equation equates the target statement to a quantity defined by the same statement, no parameter is fitted to data and then re-predicted, and no load-bearing uniqueness claim is imported from the author’s own prior work. The construction is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The result rests on three external theorems (Powers simplicity, Haagerup–Thorbjørnsen non-group Ext, Kavruk dual-exactness link) plus standard dilation theory; the only paper-specific invention is the concrete hyperrigid triple (1,D,K) engineered so that spectral projections of D together with the partial-isometry graph of K recover all matrix units and free generators.

axioms (5)
  • domain assumption C*_r(F2) is simple (Powers)
    Used in §2 to guarantee every unital representation into the Calkin algebra is injective.
  • domain assumption Ext(C*_r(F2)) is not a group (Haagerup–Thorbjørnsen)
    Supplies the non-invertible class [τ] that is amplified to [τ4] and blocked by hyperrigidity.
  • standard math Arveson’s extension theorem and Stinespring dilation
    Invoked to extend maps from S and to analyse dilations in the unique-extension proof.
  • domain assumption Kavruk’s theorem linking failure of lifting to non-exactness of the dual
    Cited for Corollary 4.3; not re-proved.
  • domain assumption Matrix-range duality: equal full matrix ranges iff the generator map is a complete-order isomorphism
    Used in the proof of Theorem 4.2 to convert a compact perturbation into a u.c.p. lift.
invented entities (1)
  • The concrete hyperrigid operator system S = span{1, D, K} with D = diag(1,2,3,4) and the indicated 4×4 self-adjoint matrix K whose off-diagonal entries are the free unitaries u,v and the unit no independent evidence
    purpose: To generate the full M4(C*_r(F2)) while possessing the unique-extension property, thereby transferring the Ext obstruction to a three-dimensional Calkin subsystem.
    The matrices are hand-crafted so that spectral projections of D together with the partial-isometry graph of K recover all matrix units and free generators; no independent existence claim is made outside this construction.

pith-pipeline@v1.1.0-grok45 · 12612 in / 2650 out tokens · 30582 ms · 2026-07-14T16:22:20.248663+00:00 · methodology

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read the original abstract

Harris recently showed that a non-liftable injective representation into the Calkin algebra gives explicit four-dimensional operator systems in the Calkin algebra without the lifting property, and hence a counterexamples to the generalized Smith--Ward problem for four-dimensional operator systems. The main obstruction also appears in an earlier work by Paulsen on this problem. We isolate the relevant part of this argument and replace the four-dimensional operator system by a three-dimensional hyperrigid operator system inside a matrix amplification of \[ C_r^*(\F_2). \] The resulting Calkin subsystem is of the form span$\{1,q(D),q(K)\}$, where $D$ and $K$ are selfadjoint operators, and the identity map on this operator system has no unital completely positive lift. Equivalently, the operator $D+iK$ gives a counterexample to the Smith--Ward problem. By a result of Kavruk, the dual of this operator system fails to be exact, and hence is the first example of a three-dimensional operator system that is not exact.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Ubiquity of counterexamples to the Smith-Ward problem

    math.OA 2026-07 accept novelty 7.5

    Every finitely generated C*-algebra without LLP contains a hyperrigid three-dimensional operator subsystem without LP (and often without exactness), giving ubiquitous Smith-Ward counterexamples and a 3D nuclearity detector.

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Works this paper leans on

13 extracted references · 1 linked inside Pith · cited by 1 Pith paper

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