Conformal nets, maximal temperature and models from free probability

We consider conformal nets on S-1 of von Neumann algebras, acting on the fu ll Fock space, arising in Free Probability. These models are twisted local, but non-local. We extend to the non-local case the general analysis of the modular structure. The local algebras turn out to be III1-factors associat ed with free groups. We use our setup to show examples exhibiting arbitrari ly large maximal temperatures, but failing to satisfy the split property, t hen clarifying the relation between the latter property and the trace class conditions on e(-betaL), where L is the conformal Hamiltonian.