Uniqueness results for the moonshine vertex operator algebra

It is proved that the vertex operator algebra V is isomorphic to the moonshine VOA 0 of Frenkel-Lepowsky-Meurman if it satisfies conditions (a,b,c,d) or (a',b,c,d). These conditions are: (a) V is the only irreducible module for itself and V is C-2-cofinite; (a') dim V <= dim V-n for n >= 3; (b) the central charge is 24; (c) V-1 = 0; (d) V-2 (under the first product on V) is isomorphic to the Griess algebra. Our two main theorems therefore establish weak versions of the FLM uniqueness conjecture for the moonshine vertex operator algebra. We believe that these are the first such results.