Normal Hilbert modules over the ball algebra A(B)

• Autorzy: Kunyu Guo
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 135
• Numer: 1
• Strony: 1-12

Daty

wydano

1999

otrzymano

1996-06-20

poprawiono

1997-12-09

poprawiono

1998-11-23

Twórcy

autor

Kunyu Guo

• Department of Mathematics, Fudan University, Shanghai 200433, P.R. China

Bibliografia

• [1] J. F. Carlson and D. N. Clark, Cohomology and extensions of Hilbert modules, J. Funct. Anal. 128 (1995), 278-306.
• [2] J. F. Carlson, D. N. Clark, C. Foiaş and J. P. Williams, Projective Hilbert A(D)-modules, New York J. Math. 1 (1994), 26-38.
• [3] X. M. Chen and R. G. Douglas, Rigidity of Hardy submodules on the unit ball, Houston J. Math. 18 (1992), 117-125.
• [4] X. M. Chen and K. Y. Guo, Cohomology and extensions of hypo-Shilov modules over the unit modulus algebras, J. Operator Theory, to appear.
• [5] J. B. Cole and T. W. Gamelin, Tight uniform algebras and algebras of analytic functions, J. Funct. Anal. 46 (1982), 158-220.
• [6] R. G. Douglas and V. I. Paulsen, Hilbert Modules over Function Algebras, Longman Sci. Tech., New York, 1989.
• [7] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer, New York, 1970.
• [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.
• [9] S. G. Krantz, Function Theory of Several Complex Variables, Wiley, New York, 1982.
• [10] E. Løw, Inner functions and boundary values in $H^∞(Ω)$ and in smoothly bounded pseudoconvex domains, Math. Z. 185 (1984), 191-210.
• [11] A. T. Paterson, Amenability, Math. Surveys Monographs 28, Amer. Math. Soc., 1988.
• [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.