C₂ cofiniteness of vertex operator algebra V_L^+ when L is a rank one lattice

Let L be a rank one positive definite even lattice. We prove that a vertex operator algebra (VOA) V_L^+ satisfies the C_2 condition. Here, V_L^+ is a fixed point sub-VOA of the VOA V_L associated with the automorphism lifted from the -1 isometry of L.