ABSTRACT
Let H be a Hilbert space and A a function algebra. We say that H is a Hilbert module over A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2464.tif"/> if there is a multiplication (a, f) → af from A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2465.tif"/> × H to H, making H into A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2466.tif"/> -module, and if, in addition, the action is jointly continuous in the sup-norm on A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2467.tif"/> and the Hilbert space norm on H. The framework of the module developed by Douglas and Paulsen was systematically presented in [DP]. In this module context, they began the study of applications of homological theory in the categories of Hilbert modules. In view of Hilbert modules, the theory of function algebras is emphasized since it plays the analogous role of ring theory in the context of algebraic modules. Therefore, in studying Hilbert modules, as in studying any algebraic structure, the standard procedure is to look at submodules and associated quotient modules. The extension problem then appears quite naturally: given two Hilbert modules H, K, what module J may be constructed with submodule H and associated quotient module K, i.e., K ≅ J/H(= J ⊖ H)? We then have a short exact sequence E : 0 → H → α J → β K → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2468.tif"/> of Hilbert A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2469.tif"/> -modules, where α, β are Hilbert module maps. Such a sequence is called an extension of K by H, or simply J is called an extension of K by H. The set of equivalence classes of extensions of K by H (this will be defined in the next section), denoted by Ext A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2470.tif"/> (K,H), may then be given a natural A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2471.tif"/> -module structure. The homological invariant Ext A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq2472.tif"/> (–, –) is important for algebraists because in general the category of all Hilbert modules over A and maps is nonabelian. For analysts, we expect Ext-groups to be a fruitful object of study and useful tool in operator theory.
