Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-09-09925-0
Published electronically: April 23, 2009
MathSciNet review: 2506455
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. J. B. Conway, J. J. Dudziak, Von Neumann operators are reflexive. J. Reine Angew. Math. 408 (1990), 34-56. MR 1058983 (91f:47003)
  • 2. M. Didas, Dual algebras generated by von Neumann $ n$-tuples over strictly pseudoconvex sets. Dissertationes Math. 425 (2004). MR 2067612 (2005d:47009)
  • 3. M. Didas, J. Eschmeier, Subnormal tuples on strictly pseudoconvex and bounded symmetric domains. Acta Sci. Math. (Szeged) 71 (2005), 691-731. MR 2206604 (2006m:47036)
  • 4. J. E. Fornaess, Embedding strictly pseudoconvex domains in convex domains. Amer. J. Math. 98 (1976), 529-569. MR 0422683 (54:10669)
  • 5. G. M. Henkin, J. Leiterer, Theory of functions on complex manifolds. Monogr. Math. 79, Birkhäuser, Basel, 1984. MR 0774049 (86a:32002)
  • 6. J. Kraus, The slice map problem for $ \sigma$-weakly closed subspaces of von Neumann algebras. Trans. Amer. Math. Soc. 279 (1983), 357-376. MR 704620 (85e:46036)
  • 7. J. Kraus, Abelian operator algebras and tensor products. J. Operator Theory 14 (1985), 391-407. MR 808298 (87f:47064)
  • 8. J. E. McCarthy, Reflexivity of subnormal operators. Pacific J. Math. 161(2) (1993), 359-370. MR 1242204 (94h:47042)
  • 9. M. Ptak, On the reflexivity of pairs of isometries and of tensor products of some operator algebras. Studia Math. 83 (1986), 47-55. MR 829898 (87f:47066)
  • 10. M. Takesaki, Theory of operator algebras, I. Springer-Verlag, Berlin-Heidelberg-New York, 2002. MR 1873025 (2002m:46083)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A15, 47B20, 47L45, 46B28, 46K50

Retrieve articles in all journals with MSC (2000): 47A15, 47B20, 47L45, 46B28, 46K50


Additional Information

Michael Didas
Affiliation: Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
Email: didas@math.uni-sb.de

DOI: https://doi.org/10.1090/S0002-9939-09-09925-0
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.

American Mathematical Society