Reduction of free independence to tensor independence

We show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators affiliated' with a given unital *-algebra and call the associated closure monotone’. Then we prove that monotone closed operators of the form $$X^{\mathrm{\prime }}=\sum _{k=1}^{\mathrm{\infty }}X(k)\bar{\otimes }{p}_k,X^{\mathrm{\prime }\mathrm{\prime }}=\sum _{k=1}^{\mathrm{\infty }}{p}_k\bar{\otimes }X(k)$$ are free with respect to a tensor product state, where $X(k)$ are tensor independent copies of a random variable $X$ and $(p_k)$ is a sequence of orthogonal projections. For unital free *-algebras, we construct a monotone closed analog of a unital *-bialgebra called a `monotone closed quantum semigroup’ which implements the additive free convolution, without using the concept of dual groups.