Journal of Operator Theory

Volume 43, Issue 1, Winter 2000 pp. 69-81.

Cohomology and extensions of Hypo-Şilov modules over unit modulus algebras

Authors: Xiaoman Chen (1), and Kunyu Guo (2)
Author institution: (1) Institute of Mathematics, Fudan University, Shanghai, 200433, P.R. China
(2) Institute of Mathematics, Fudan University, Shanghai, 200433, P.R. China

Summary: This paper is a study of cohomology and extensions of hypo-Şilov modules over unit modulus algebras. We first prove that every $C(\partial A_U)$-extension of a hypo-Şilov module, viewed as a Hilbert module over $A_U$, is projective and injective. It follows that some interesting results are derived, especially so-called ``Hom-Isomorphism" theorem. By using ``Hom-Ext'' sequences , we can compute ${\rm Ext}_{A_U}$-groups for hypo-Şilov modules and cohypo-Şilov modules. Finally, these results are applied to the discussion of rigidity and extensions of Hardy submodules over polydisk algebras.

Keywords: Extension, hypo-Şilov module, projective module, rigidity

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