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

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 $$\{T\in \mathcal{B}(H);T\mathrm{normal}\mathrm{and}\ddot{I}ƒ(T)\subset \mathrm{ex}(K)\}.$$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown.

21 pages