Unitary representations of the W3-algebra with c ≥ 2

We prove unitarity of the vacuum representation of the W_3-algebra for all values of the central charge c ≥ 2.We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the W_3-algebra in the 2 ≤ c ≤ 98 region.