Normal Hilbert modules over the ball algebra A(B)

Kunyu Guo

Studia Mathematica (1999)

  • Volume: 135, Issue: 1, page 1-12
  • ISSN: 0039-3223

Abstract

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The normal cohomology functor E x t ℵ is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of E x t ℵ -groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B) is normal projective and normal injective. It follows that there is a natural isomorphism between Hom of normal Shilov modules and that of their quotient modules, which is a new lifting theorem of normal Shilov modules. Finally, these results are applied to the discussion of rigidity and extensions of Hardy submodules over the ball algebra.

How to cite

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Guo, Kunyu. "Normal Hilbert modules over the ball algebra A(B)." Studia Mathematica 135.1 (1999): 1-12. <http://eudml.org/doc/216640>.

@article{Guo1999,
abstract = {The normal cohomology functor $Ext_ℵ$ is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of $Ext_ℵ$-groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B) is normal projective and normal injective. It follows that there is a natural isomorphism between Hom of normal Shilov modules and that of their quotient modules, which is a new lifting theorem of normal Shilov modules. Finally, these results are applied to the discussion of rigidity and extensions of Hardy submodules over the ball algebra.},
author = {Guo, Kunyu},
journal = {Studia Mathematica},
keywords = {normal cohomology functor; normal Hilbert modules; ball algebra; normal Shilov modules; quotients modules},
language = {eng},
number = {1},
pages = {1-12},
title = {Normal Hilbert modules over the ball algebra A(B)},
url = {http://eudml.org/doc/216640},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Guo, Kunyu
TI - Normal Hilbert modules over the ball algebra A(B)
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 1
SP - 1
EP - 12
AB - The normal cohomology functor $Ext_ℵ$ is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculation of $Ext_ℵ$-groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B) is normal projective and normal injective. It follows that there is a natural isomorphism between Hom of normal Shilov modules and that of their quotient modules, which is a new lifting theorem of normal Shilov modules. Finally, these results are applied to the discussion of rigidity and extensions of Hardy submodules over the ball algebra.
LA - eng
KW - normal cohomology functor; normal Hilbert modules; ball algebra; normal Shilov modules; quotients modules
UR - http://eudml.org/doc/216640
ER -

References

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  6. [6] R. G. Douglas and V. I. Paulsen, Hilbert Modules over Function Algebras, Longman Sci. Tech., New York, 1989. 
  7. [7] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer, New York, 1970. Zbl0863.18001
  8. [8] N. P. Jewell, Multiplication by the coordinate functions on the Hardy space of the unit sphere of ℂ n , Duke Math. J. 44 (1977), 839-851. Zbl0372.47016
  9. [9] S. G. Krantz, Function Theory of Several Complex Variables, Wiley, New York, 1982. Zbl0471.32008
  10. [10] E. Løw, Inner functions and boundary values in H ∞ ( Ω ) and in smoothly bounded pseudoconvex domains, Math. Z. 185 (1984), 191-210. Zbl0526.32017
  11. [11] A. T. Paterson, Amenability, Math. Surveys Monographs 28, Amer. Math. Soc., 1988. 
  12. [12] W. Rudin, New constructions of functions holomorphic in the unit ball of C n , CBMS Regional Conf. Ser. in Math. 63, Amer. Math. Soc., 1986. 

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