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Abstract

Let $D \Subset X$ denote a relatively compact strictly pseudoconvex open subset of a Stein submanifold $X\subset {\sym C}^n$ and let $H$ be a separable complex Hilbert space. By a von Neumann $n$-tuple of class ${\sym A}$ over $D$ we mean a commuting $n$-tuple of operators $T \in L(H)^n$ possessing an isometric and weak$^*$ continuous $H^\infty(D)$-functional calculus as well as a $\partial D$-unitary dilation. The aim of this paper is to present an introduction to the structure theory of von Neumann $n$-tuples of class ${\sym A}$ over $D$ including the necessary function- and measure-theoretical background. Our main result will be a chain of equivalent conditions characterizing those von Neumann $n$-tuples of class ${\sym A}$ over $D$ which satisfy the factorization property ${\sym A}_{1,\aleph_0}$. The dual algebra generated by each such tuple is shown to be super-reflexive. As a consequence we deduce that each subnormal tuple possessing an isometric and weak$^*$ continuous $H^\infty(D)$-functional calculus and each subnormal tuple with dominating Taylor spectrum in $D$ is reflexive.