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For the sake of providing insight into the use of nonstandard techniques à la A. Robinson and into Luxemburg's nonstandard hull construction, we first present nonstandard proofs of some known results about C * -algebras. Then we introduce extensions of the nonstandard hull construction to noncommutative probability spaces and noncommutative stochastic processes. In the framework of internal noncommutative probability spaces, we investigate properties like freeness and convergence in distribution and their preservation by the nonstandard hull construction. We obtain a nonstandard characterization of the freeness property. Eventually we provide a nonstandard characterization of the property of equivalence for a suitable class of noncommutative stochastic processes and we study the behaviour of the latter property with respect to the nonstandard hull construction. [ABSTRACT FROM AUTHOR]