$C^{\ast }$-supports and abnormalities of operator systems

Let $S$ be a concrete operator system represented on some Hilbert space $H$. A $C^{\ast }$-support of $S$ is the $C^{\ast }$-algebra generated (via the Choi–Effros product) by $S$ inside an injective operator system acting on $H$. By leveraging Hamana’s theory, we show that such a $C^{\ast }$-support is unique precisely when $C^{\ast }(S)$ is contained in every copy of the injective envelope of $S$ that acts on $H$. Further, we demonstrate how the uniqueness of certain $C^{\ast }$-supports can be used to give new characterizations of the unique extension property for $\ast$-representations, as well as the hyperrigidity of $S$. In another direction, we utilize the collection of all $C^{\ast }$-supports of $S$ to describe the subspace generated by the so-called abnormalities of $S$, thereby complementing a result of Kakariadis.