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On the slice map problem for $ H^\infty(\Omega)$ and the reflexivity of tensor products

Author: Michael Didas
Journal: Proc. Amer. Math. Soc. 137 (2009), 2969-2978
MSC (2000): Primary 47A15, 47B20, 47L45; Secondary 46B28, 46K50
Published electronically: April 23, 2009
MathSciNet review: 2506455
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Abstract: Let $ \Omega \subset \mathbb{C}^n$ be a bounded convex or strictly pseudoconvex open subset. Given a separable Hilbert space $ K$ and a weak$ ^*$ closed subspace $ \mathcal{T} \subset B(K)$, we show that the space $ H^\infty(\Omega, \mathcal{T})$ of all bounded holomorphic $ \mathcal{T}$-valued functions on $ \Omega$ possesses the tensor product representation $ H^\infty(\Omega, \mathcal{T}) = H^\infty(\Omega){\overline{\otimes}}\mathcal{T}$ with respect to the normal spatial tensor product. As a consequence we deduce that $ H^\infty(\Omega)$ has property $ S_\sigma$. This implies that, if $ S\in B(H)^n$ is a subnormal tuple of class $ \mathbb{A}$ on a strictly pseudoconvex or bounded symmetric domain and $ T \in B(K)^m$ is a commuting tuple satisfying AlgLat$ (T) = \mathcal{A}_T$ (where $ \mathcal{A}_T$ denotes the unital dual operator algebra generated by $ T$), then the tensor product tuple $ (S\otimes 1, 1 \otimes T)$ is reflexive.

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Additional Information

Michael Didas
Affiliation: Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany

Keywords: Property $S_\sigma $, strictly pseudoconvex domains, subnormal tuples, tensor products
Received by editor(s): February 16, 2007
Received by editor(s) in revised form: August 26, 2007
Published electronically: April 23, 2009
Dedicated: This paper is dedicated to Christine and Tim
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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