The Hyperrigidity Conjecture for compact convex sets in $\mathbb{R}^2$

TL;DR

This paper proves the hyperrigidity of the operator system of continuous affine functions on any compact convex set in b2, linking geometric properties to operator algebraic rigidity and generalizing prior results.

Contribution

It establishes hyperrigidity for all compact convex sets in b2, extending previous work and connecting geometric features with operator topologies.

Findings

Operator system A(K) is hyperrigid in C( ext{ex}(K)) for all compact convex K in b2.

Weak and strong operator topologies coincide on certain normal operators with spectra in ext{ex}(K).

The approach generalizes previous results by Brown using geometric properties of K.

Abstract

We prove that for every compact, convex subset $K\subset\mathbb{R}^2$ the operator system $A(K)$, consisting of all continuous affine functions on $K$, is hyperrigid in the C*-algebra $C(\mathrm{ex}(K))$. In particular, this result implies that the weak and strong operator topologies coincide on the set $$ \{ T\in\mathcal{B}(H);\ T\ \mathrm{normal}\ \mathrm{and}\ \sigma(T)\subset \mathrm{ex}(K) \}. $$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown.