Quantum operations on conformal nets

On a conformal net $\mathcal{A}$, one can consider collections of unital completely positive maps on each local algebra $\mathcal{A}(I)$, subject to natural compatibility, vacuum preserving and covariance conditions. We call \emph{quantum operations} $\mathcal{A}$ the subset extreme such maps. The usual automorphisms (the invertible *-algebra morphisms) are examples quantum operations, we show that fixed point subnet under all operations is Virasoro generated by stress-energy tensor $\mathcal{A}$. Furthermore, every irreducible $\mathcal{B}\subset\mathcal{A}$ points operations. When discrete (or with finite Jones index), set leave $\mathcal{B}$ elementwise has naturally structure compact finite) hypergroup, thus extending some results [Bis17]. Under same assumptions, provide Galois correspondence between intermediate nets closed subhypergroups. In particular, in one-to-one subfactors, result Longo index/completely rational setting [Lon03].