Hyperrigid operator systems and Hilbert modules

Abstract

It is shown that, for an operator algebra $A$, the operator system $S=A+A^{\ast }$ in the $C^{\ast }$-algebra $C^{\ast }\left(S\right)$, and any representation $\rho$ of $C^{\ast }\left(S\right)$ on a Hilbert space $\mathcal{H}$, the restriction ${\rho }_{{|}_S}$ has a unique extension property if and only if the Hilbert module $\mathcal{H}$ over $A$ is both orthogonally projective and orthogonally injective. As a corollary we deduce that, when $S$ is separable, the hyperrigidity of $S$ is equivalent to the Hilbert modules over $A$ being both orthogonally projective and orthogonally injective.