The Hyperrigidity Conjecture for Spectrahedra

Abstract: We show that if K is a compact spectrahedron whose set of extreme points is closed, then the operator system of continuous affine functions on K is hyperrigid in the C*-algebra C(ex(K)).

• This paper proves that for closed extreme points of compact spectrahedra, the corresponding operator system of affine functions is hyperrigid, validating Arveson's hyperrigidity conjecture in this setting.
• The paper leverages Perron-Frobenius estimates and properties of spectrahedra geometry to show that the operator system A(K) is hyperrigid in C(√{}¤X(K) citiesiguta='t esc(√{ex}{K}))' ,this is a criterion for hyperrigidity in C^(S)
• The result significantly extends known instances under the large class of commutative models where the conjecture stands true.

This paper by Marcel Scherer establishes hyperrigidity of function systems associated with compact spectrahedra whose set of extreme points is closed. Specifically, it proves that if $K \subset \mathbb{R}^k$ is a compact spectrahedron with $\textup{ex}(K)$ closed, then $A(K)$, the operator system of continuous affine functions on $K$, is hyperrigid in $C(\textup{ex}(K))$ (2601.16075). The result is significant because Arveson's hyperrigidity conjecture is now known to fail in general, but all known counterexamples live in non-commutative $C^*$-algebras; this paper confirms that the conjecture holds for spectrahedral function systems, a large and natural class of commutative examples.

Background: hyperrigidity and the conjecture

An operator system $S$ is hyperrigid if every unital $*$-homomorphism of $C^*(S)$ restricts to a u.c.p.\ map on $S$ with the unique extension property (u.e.p.). Arveson's hyperrigidity conjecture asserts that it suffices to check irreducible representations. Recent work has produced counterexamples in full generality (Bilich et al., 2024, Bilich, 2024, Scherer, 14 Feb 2025, Clouâtre, 23 Sep 2025), but each has non-commutative $\textup{ex}(K)$0. This motivates the commutative setting, where $\textup{ex}(K)$1 and every irreducible representation is an evaluation map.

For a function system $\textup{ex}(K)$2 with state space $\textup{ex}(K)$3, the canonical map $\textup{ex}(K)$4 embeds $\textup{ex}(K)$5 completely isometrically with dense image into $\textup{ex}(K)$6. Two structural facts drive the paper's strategy:

• $\textup{ex}(K)$7 is hyperrigid in its $\textup{ex}(K)$8-envelope if and only if $\textup{ex}(K)$9 is hyperrigid in $A(K)$0.
• Every irreducible representation restricts to a map with the u.e.p.\ if and only if $A(K)$1.

Thus, under the closedness assumption on $A(K)$2, the irreducible-representation criterion is automatically satisfied, and the task is to prove genuine hyperrigidity. Prior results along these lines include hyperrigidity of $A(K)$3 for compact convex $A(K)$4 (Scherer, 2024), the dilation-order framework of Davidson–Kennedy [107774-related work cited as DaKe], and singly generated commutative examples (Pietrzycki et al., 2024). Spectrahedra constitute a strictly larger class than planar convex sets, so the theorem substantially extends the known territory.

Perron–Frobenius estimates

The proof requires quantitative control of Perron eigenvalues and eigenvectors of selfadjoint matrices with strictly positive entries bounded between constants $A(K)$5. The key lemma shows that the normalized Perron eigenvector $A(K)$6 satisfies $A(K)$7 for every coordinate, that $A(K)$8, and consequently that every entry of the rank-one matrix $A(K)$9 is at least $K$0. These uniform lower bounds are essential later, since they guarantee strict positivity of entries of certain rank-one perturbations independent of dimension or location — a point on which the entire argument hinges.

Geometry of spectrahedra

After symmetrizing the defining pencil (replacing a selfadjoint pencil $K$1 by an equivalent symmetric one via the real/imaginary block construction), the paper analyzes the stratification of the boundary by kernel dimension:

$K$2

Three geometric lemmas form the backbone:

• Extreme points are isolated in their stratum: for $K$3 with $K$4 the projection onto $K$5, the set $K$6 equals $K$7. The proof identifies this set with the smallest face containing $K$8, using the classification of faces of the positive semidefinite cone and the fact that the affine map $K$9 carries faces of $C(\textup{ex}(K))$0 to faces of $C(\textup{ex}(K))$1.
• Strata are locally closed: each $C(\textup{ex}(K))$2 is the intersection of the closed set $C(\textup{ex}(K))$3 with the complement of $C(\textup{ex}(K))$4.
• Local continuous kernel sections exist: near any extreme point $C(\textup{ex}(K))$5 and for any $C(\textup{ex}(K))$6, there is a closed subset $C(\textup{ex}(K))$7 containing a neighborhood of $C(\textup{ex}(K))$8 within $C(\textup{ex}(K))$9, and a continuous normalized section $C^*$0 of the kernel bundle such that $C^*$1, where $C^*$2.

The continuity of $C^*$3 is obtained via Riesz spectral projections, which depend continuously on $C^*$4 once the zero eigenvalue is separated from the rest of the spectrum. The paper notes explicitly that a globally continuous choice of $C^*$5 on all of $C^*$6 cannot be expected in general; the local construction is what makes the argument work.

The core estimate

The central technical result (Lemma "Main") assumes $C^*$7 and produces, for disjoint closed $C^*$8 and an extreme point $C^*$9 satisfying the positivity condition above, a neighborhood $S$0 of $S$1 with

$S$2

whenever $S$3 is a unital $S$4-homomorphism and $S$5 a u.c.p.\ map agreeing with $S$6 on $S$7. The mechanism is a five-step argument:

1. Cover $S$8 by finitely many sets on which the diagonal kernels $S$9 are uniformly small ($*$0).
2. Encode the kernel in the affine matrix-valued map $*$1, which is positive over $*$2.
3. Split $*$3 into negative and positive spectral parts; on a neighborhood of $*$4, the negative part contributes at most $*$5 entrywise, while the positive part $*$6 has entries in $*$7.
4. Express $*$8 as a convex combination $*$9 with $C^*(S)$0 and $C^*(S)$1, producing a ball $C^*(S)$2.
5. Apply Reams' theorem on Hadamard inverses [Rea]: since $C^*(S)$3 has strictly positive entries and only one positive eigenvalue, its Hadamard inverse $C^*(S)$4 is positive semidefinite, whence $C^*(S)$5 and ultimately $C^*(S)$6.

Compressing by the projections $C^*(S)$7 then yields the norm bound $C^*(S)$8, and since $C^*(S)$9 and $S$0 are independent of $S$1, letting $S$2 gives vanishing. This quantitative entrywise bookkeeping — the interplay between the Perron lower bound $S$3 and the spectral-splitting constant $S$4 — is the analytic heart of the paper.

A general separation principle

The final step is isolated as a theorem of independent interest, valid for arbitrary compact metrizable $S$5: if $S$6 is a unital $S$7-homomorphism, $S$8 a u.c.p.\ map, and for every pair of distinct points there are disjoint neighborhoods $S$9 with $\textup{ex}(K)$00, then $\textup{ex}(K)$01 on all of $\textup{ex}(K)$02. The proof first propagates the vanishing to all pairs of disjoint open sets using Brown's convergence theorem [Br] and a Lindelöf exhaustion, then upgrades to equality on characteristic functions of open sets via WOT/SOT limiting arguments, and finally invokes the Dynkin $\textup{ex}(K)$03-$\textup{ex}(K)$04 theorem to conclude equality on all Borel sets. Since this step uses no spectrahedral structure, it applies to hyperrigidity problems for other function systems as well.

Assembly of the main theorem

For $\textup{ex}(K)$05 with nonempty interior, fix distinct extreme points $\textup{ex}(K)$06. One covers $\textup{ex}(K)$07 for each stratum $\textup{ex}(K)$08 by countably many neighborhoods obtained from the local-section lemma and the core estimate, takes SOT limits using Brown's theorem, and sums over strata to obtain $\textup{ex}(K)$09. Points outside $\textup{ex}(K)$10 are handled separately using the closedness of $\textup{ex}(K)$11, which supplies neighborhoods avoiding the extreme set entirely. The separation principle then forces $\textup{ex}(K)$12.

If $\textup{ex}(K)$13, the paper reduces to the full-dimensional case by restricting to the affine hull: an affine homeomorphism $\textup{ex}(K)$14 transports $\textup{ex}(K)$15 to a compact spectrahedron $\textup{ex}(K)$16 with nonempty interior, preserves extreme points, and induces a unital complete order isomorphism $\textup{ex}(K)$17. Singletons are trivial. This reduction is clean but relies on compactness being preserved under the restriction of the pencil to the affine hull, which holds because $\textup{ex}(K)$18 is injective on the relevant space.

Limitations and open questions

Two restrictions bound the scope of the theorem. First, the hypothesis that $\textup{ex}(K)$19 is closed is not automatic for spectrahedra in dimension three and higher; the theorem says nothing about compact spectrahedra with non-closed extreme sets, where even the u.e.p.\ criterion for irreducible representations fails. Second, the standing compactness assumption excludes unbounded spectrahedra, which arise naturally in semidefinite programming. Whether the methods extend to non-compact spectrahedra, or whether the closed-extreme-points hypothesis can be weakened, remains open. It would also be natural to ask whether the Hadamard-product technique combined with the separation principle applies to broader classes of semialgebraic convex sets beyond those admitting a linear matrix inequality representation.

Conclusion

The paper proves that compact spectrahedra with closed extreme-point sets yield hyperrigid function systems, thereby verifying Arveson's hyperrigidity conjecture for a substantial commutative class consistent with the pattern that all counterexamples are non-commutative. The proof combines a quantitative Perron–Frobenius analysis, local continuous sections of the kernel bundle of the defining pencil, a Hadamard-inverse domination argument, and a measure-theoretic separation principle stated in sufficient generality for reuse elsewhere. The result places spectrahedra alongside planar convex sets and singly generated systems among the known instances where the conjecture holds in the commutative setting.