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Let D ⋐ X denote a relatively compact strictly pseudoconvex open subset of a Stein submanifold X ⊂ ℂⁿ and let H be a separable complex Hilbert space. By a von Neumann n-tuple of class over D we mean a commuting n-tuple of operators T ∈ L(H)ⁿ possessing an isometric and weak* continuous $H^{\infty }\left(D\right)$-functional calculus as well as a ∂D-unitary dilation. The aim of this paper is to present an introduction to the structure theory of von Neumann n-tuples of class over D including the necessary function- and measure-theoretical...