The Automorphism Group of the Vertex Operator Algebra $V_L^{+}$ for an even lattice $L$ without roots

The automorphism group of the vertex operator algebra $V_L^{+}$ is studied by using its action on isomorphism classes of irreducible $V_L^{+}$-modules. In particular, the shape of the automorphism group of $V_L^{+}$ is determined when $L$ is isomorphic to an even unimodular lattice without roots, $\sqrt{2}R$ for an irreducible root lattice $R$ of type $ADE$ and the Barnes-Wall lattice of rank 16.