C₁-cofiniteness and Fusion Products of Vertex Operator Algebras

Let V be a vertex operator algebra. We prove that if U and W are C_1-cofinite {\mathbb N}-gradable V-modules, then a fusion product U\boxtimes W is well-defined and also a C_1-cofinite {\mathbb N}-gradable V-module, where the fusion product is defined by (logarithmic) intertwining operators. This is also true for C_2-cofinite {\mathbb N}-gradable modules.