The Hyperrigidity Conjecture for Spectrahedra

TL;DR

The paper proves that for a compact spectrahedron $K$ with closed $ex(K)$, the function system $A(K)$ is hyperrigid in $C(ex(K))$. The approach translates hyperrigidity to positive kernels derived from a symmetric pencil $Q$, constructs a matrix-valued map in $M_n(A(K))$, and leverages Perron–Frobenius bounds and a Hadamard product domination to enforce the unique extension property. A general separation principle completes the argument, showing that any u.c.p. map matching a $*$-homomorphism on $A(K)$ extends to the whole $C(ex(K))$. The results extend known hyperrigid examples to spectrahedra, linking convex geometry and operator-algebraic dilation theory in a commutative setting.

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)).

Theorems & Definitions (20)

• Conjecture : Hyperrigidity Conjecture
• Proposition
• Proposition
• Theorem
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof