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The lattice vertex operator algebra V L  associated to a positive definite even lattice L has an automorphism of order 2 lifted from −1-isometry of L. The fixed point set V L + of V L  for the automorphism is naturally a vertex operator algebra. We prove that any ℤ≥0-graded weak V L +-module is completely reducible.

Rationality of the vertex operator algebra VL+ for a positive definite even lattice L