C⁎-supports and abnormalities of operator systems

Abstract

Let S be a concrete operator system represented on some Hilbert space H. A $C^{⁎}$-support of S is the $C^{⁎}$-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 $C^{⁎}$-support is unique precisely when $C^{⁎}\left(S\right)$ (the $C^{⁎}$-algebra generated in $B\left(H\right)$ with the usual product) is contained in every copy of the injective envelope of S that acts on H. Further, we demonstrate how the uniqueness of certain $C^{⁎}$-supports can be used to give new characterizations of the unique extension property for ⁎-representations, as well as the hyperrigidity of S. In another direction, we utilize the collection of all $C^{⁎}$-supports of S to describe the subspace generated by the so-called abnormalities of S, thereby complementing an earlier result of Kakariadis.