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Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets

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Didas M.

Instytut Matematyczny Polskiej Akademii Nauk

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 𝔸 over D we mean a commuting n-tuple of operators T ∈ L(H)ⁿ possessing an isometric and weak* continuous $H^{∞}(D)$-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 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 𝔸 over D which satisfy the factorization property $𝔸_{1,ℵ₀}$. 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 $H^{∞}(D)$-functional calculus and each subnormal tuple with dominating Taylor spectrum in D is reflexive.

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1 Didas M.

1 2004

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Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets