A genus zero functorial CFT associated to the vacuum sector of a conformal net

Starting from an arbitrary conformal net, we construct a genus zero functorial conformal field theory whose Hilbert space is the vacuum sector of the net. Specifically, we construct an algebra for the operad of little discs and conformal embeddings with values in the category of Hilbert spaces and bounded linear maps. We work with closed discs, and our conformal embeddings explicitly allow the image of an incoming disc to overlap with the boundary of the outgoing disc. As an application, we show that all conformal nets satisfy the trace class condition: if $L_0$ is the conformal Hamiltonian of the net, then the operators $r^{L_0}$ are trace class whenever $0 \le r < 1$. In particular, the $L_0$-eigenspaces of a conformal net are automatically finite-dimensional.