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This is the sixth part in a series of papers in which we introduce anddevelop a natural, general tensor category theory for suitable modulecategories for a vertex (operator) algebra. In this paper (Part VI), weconstruct the appropriate natural associativity isomorphisms between tripletensor product functors. In fact, we establish a "logarithmic operator productexpansion" theorem for logarithmic intertwining operators. In this part, agreat deal of analytic reasoning is needed; the statements of the main theoremsthemselves involve convergence assertions.

Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms