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 -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 background. Our main result will be a chain of equivalent conditions characterizing those von Neumann n-tuples of class over D which satisfy the factorization property . 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 -functional calculus and each subnormal tuple with dominating Taylor spectrum in D is reflexive.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm425-0-1,
author = {Michael Didas},
title = {Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets},
series = {GDML\_Books},
year = {2004},
zbl = {1065.47002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm425-0-1}
}
Michael Didas. Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets. GDML_Books (2004), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm425-0-1/